id sid tid token lemma pos cana-282 1 1 communications communication NOUN cana-282 1 2 on on ADP cana-282 1 3 applied apply VERB cana-282 1 4 nonlinear nonlinear ADJ cana-282 1 5 analysis analysis NOUN cana-282 1 6 issn issn NOUN cana-282 1 7 : : PUNCT cana-282 1 8 1074 1074 NUM cana-282 1 9 - - PUNCT cana-282 1 10 133x 133x NUM cana-282 1 11 vol vol NOUN cana-282 1 12 30 30 NUM cana-282 1 13 no no NOUN cana-282 1 14 . . NOUN cana-282 1 15 4 4 NUM cana-282 1 16 ( ( PUNCT cana-282 1 17 2023 2023 NUM cana-282 1 18 ) ) PUNCT cana-282 1 19 39 39 NUM cana-282 1 20 https://internationalpubls.com https://internationalpubls.com X cana-282 1 21 a a DET cana-282 1 22 modified modify VERB cana-282 1 23 trapezoidal trapezoidal ADJ cana-282 1 24 technique technique NOUN cana-282 1 25 with with ADP cana-282 1 26 improved improved ADJ cana-282 1 27 accuracy accuracy NOUN cana-282 1 28 of of ADP cana-282 1 29 nonlinear nonlinear ADJ cana-282 1 30 algebraic algebraic ADJ cana-282 1 31 and and CCONJ cana-282 1 32 transcendental transcendental ADJ cana-282 1 33 equations equation NOUN cana-282 1 34 inderjeet1 inderjeet1 PROPN cana-282 1 35 , , PUNCT cana-282 1 36 rashmi rashmi PROPN cana-282 1 37 bhardwaj2 bhardwaj2 PROPN cana-282 1 38 * * PROPN cana-282 1 39 1research 1research NUM cana-282 1 40 scholar scholar NOUN cana-282 1 41 , , PUNCT cana-282 1 42 university university NOUN cana-282 1 43 school school NOUN cana-282 1 44 of of ADP cana-282 1 45 basic basic ADJ cana-282 1 46 and and CCONJ cana-282 1 47 applied applied ADJ cana-282 1 48 sciences science NOUN cana-282 1 49 , , PUNCT cana-282 1 50 department department NOUN cana-282 1 51 of of ADP cana-282 1 52 mathematics mathematic NOUN cana-282 1 53 , , PUNCT cana-282 1 54 guru guru NOUN cana-282 1 55 gobind gobind PROPN cana-282 1 56 singh singh PROPN cana-282 1 57 indraprastha indraprastha PROPN cana-282 1 58 university university PROPN cana-282 1 59 , , PUNCT cana-282 1 60 dwarka dwarka NOUN cana-282 1 61 , , PUNCT cana-282 1 62 new new ADJ cana-282 1 63 delhi delhi PROPN cana-282 1 64 , , PUNCT cana-282 1 65 india india PROPN cana-282 1 66 . . PROPN cana-282 1 67 2 2 NUM cana-282 1 68 faculty faculty NOUN cana-282 1 69 of of ADP cana-282 1 70 university university NOUN cana-282 1 71 school school NOUN cana-282 1 72 of of ADP cana-282 1 73 basic basic ADJ cana-282 1 74 and and CCONJ cana-282 1 75 applied applied ADJ cana-282 1 76 sciences science NOUN cana-282 1 77 , , PUNCT cana-282 1 78 head head NOUN cana-282 1 79 of of ADP cana-282 1 80 department department NOUN cana-282 1 81 of of ADP cana-282 1 82 mathematics mathematic NOUN cana-282 1 83 , , PUNCT cana-282 1 84 guru guru NOUN cana-282 1 85 gobind gobind PROPN cana-282 1 86 singh singh PROPN cana-282 1 87 indraprastha indraprastha PROPN cana-282 1 88 university university PROPN cana-282 1 89 , , PUNCT cana-282 1 90 dwarka dwarka NOUN cana-282 1 91 , , PUNCT cana-282 1 92 new new ADJ cana-282 1 93 delhi delhi PROPN cana-282 1 94 , , PUNCT cana-282 1 95 india india PROPN cana-282 1 96 . . PUNCT cana-282 2 1 1email 1email NUM cana-282 2 2 i i SYM cana-282 2 3 d d NOUN cana-282 2 4 : : PUNCT cana-282 2 5 yadavinderjeet386@gmail.com yadavinderjeet386@gmail.com X cana-282 3 1 * * PUNCT cana-282 3 2 corresponding correspond VERB cana-282 3 3 author author NOUN cana-282 3 4 email email NOUN cana-282 3 5 i i PROPN cana-282 3 6 d d PROPN cana-282 3 7 : : PUNCT cana-282 3 8 rashmib@ipu.ac.in rashmib@ipu.ac.in NOUN cana-282 3 9 article article NOUN cana-282 3 10 history history NOUN cana-282 3 11 : : PUNCT cana-282 3 12 received receive VERB cana-282 3 13 : : PUNCT cana-282 3 14 12 12 NUM cana-282 3 15 - - SYM cana-282 3 16 08 08 NUM cana-282 3 17 - - PUNCT cana-282 3 18 2023 2023 NUM cana-282 3 19 revised revise VERB cana-282 3 20 : : PUNCT cana-282 3 21 22 22 NUM cana-282 3 22 - - SYM cana-282 3 23 09 09 NUM cana-282 3 24 - - PUNCT cana-282 3 25 2023 2023 NUM cana-282 3 26 accepted accept VERB cana-282 3 27 : : PUNCT cana-282 3 28 15 15 NUM cana-282 3 29 - - SYM cana-282 3 30 10 10 NUM cana-282 3 31 - - PUNCT cana-282 3 32 2023 2023 NUM cana-282 3 33 abstract abstract NOUN cana-282 3 34 : : PUNCT cana-282 3 35 for for ADP cana-282 3 36 the the DET cana-282 3 37 purpose purpose NOUN cana-282 3 38 of of ADP cana-282 3 39 evaluating evaluate VERB cana-282 3 40 numerical numerical ADJ cana-282 3 41 integration integration NOUN cana-282 3 42 , , PUNCT cana-282 3 43 a a DET cana-282 3 44 modified modify VERB cana-282 3 45 trapezoidal trapezoidal ADJ cana-282 3 46 rule rule NOUN cana-282 3 47 is be AUX cana-282 3 48 presented present VERB cana-282 3 49 in in ADP cana-282 3 50 this this DET cana-282 3 51 paper paper NOUN cana-282 3 52 ; ; PUNCT cana-282 3 53 the the DET cana-282 3 54 proposed propose VERB cana-282 3 55 method method NOUN cana-282 3 56 appears appear VERB cana-282 3 57 to to PART cana-282 3 58 be be AUX cana-282 3 59 an an DET cana-282 3 60 effective effective ADJ cana-282 3 61 modification modification NOUN cana-282 3 62 of of ADP cana-282 3 63 the the DET cana-282 3 64 trapezoidal trapezoidal ADJ cana-282 3 65 rule rule NOUN cana-282 3 66 , , PUNCT cana-282 3 67 which which PRON cana-282 3 68 consists consist VERB cana-282 3 69 of of ADP cana-282 3 70 the the DET cana-282 3 71 arithmetic arithmetic ADJ cana-282 3 72 mean mean NOUN cana-282 3 73 of of ADP cana-282 3 74 subintervals subinterval NOUN cana-282 3 75 , , PUNCT cana-282 3 76 while while SCONJ cana-282 3 77 the the DET cana-282 3 78 proposed propose VERB cana-282 3 79 method method NOUN cana-282 3 80 uses use VERB cana-282 3 81 the the DET cana-282 3 82 arithmetic arithmetic ADJ cana-282 3 83 mean mean NOUN cana-282 3 84 and and CCONJ cana-282 3 85 heronian heronian ADJ cana-282 3 86 mean mean NOUN cana-282 3 87 at at ADP cana-282 3 88 the the DET cana-282 3 89 subintervals subinterval NOUN cana-282 3 90 . . PUNCT cana-282 4 1 compared compare VERB cana-282 4 2 to to ADP cana-282 4 3 the the DET cana-282 4 4 original original ADJ cana-282 4 5 trapezoidal trapezoidal ADJ cana-282 4 6 rule rule NOUN cana-282 4 7 , , PUNCT cana-282 4 8 this this DET cana-282 4 9 rule rule NOUN cana-282 4 10 has have VERB cana-282 4 11 a a DET cana-282 4 12 higher high ADJ cana-282 4 13 accuracy accuracy NOUN cana-282 4 14 . . PUNCT cana-282 5 1 utilizing utilize VERB cana-282 5 2 numerical numerical ADJ cana-282 5 3 experiments experiment NOUN cana-282 5 4 on on ADP cana-282 5 5 the the DET cana-282 5 6 basis basis NOUN cana-282 5 7 of of ADP cana-282 5 8 local local ADJ cana-282 5 9 truncation truncation NOUN cana-282 5 10 error error NOUN cana-282 5 11 , , PUNCT cana-282 5 12 the the DET cana-282 5 13 modified modified ADJ cana-282 5 14 and and CCONJ cana-282 5 15 original original ADJ cana-282 5 16 trapezoidal trapezoidal ADJ cana-282 5 17 rules rule NOUN cana-282 5 18 are be AUX cana-282 5 19 compared compare VERB cana-282 5 20 keywords keyword NOUN cana-282 5 21 : : PUNCT cana-282 5 22 arithmetic arithmetic ADJ cana-282 5 23 mean mean NOUN cana-282 5 24 , , PUNCT cana-282 5 25 heronian heronian ADJ cana-282 5 26 mean mean NOUN cana-282 5 27 , , PUNCT cana-282 5 28 modified modified ADJ cana-282 5 29 trapezoidal trapezoidal ADJ cana-282 5 30 technique technique NOUN cana-282 5 31 , , PUNCT cana-282 5 32 numerical numerical ADJ cana-282 5 33 examples example NOUN cana-282 5 34 , , PUNCT cana-282 5 35 accuracy accuracy NOUN cana-282 5 36 , , PUNCT cana-282 5 37 and and CCONJ cana-282 5 38 numerical numerical ADJ cana-282 5 39 integration integration NOUN cana-282 5 40 . . PUNCT cana-282 6 1 1 1 X cana-282 6 2 . . X cana-282 6 3 introduction introduction NOUN cana-282 6 4 integration integration NOUN cana-282 6 5 , , PUNCT cana-282 6 6 which which PRON cana-282 6 7 is be AUX cana-282 6 8 the the DET cana-282 6 9 method method NOUN cana-282 6 10 for for ADP cana-282 6 11 determining determine VERB cana-282 6 12 the the DET cana-282 6 13 area area NOUN cana-282 6 14 that that PRON cana-282 6 15 a a DET cana-282 6 16 function function NOUN cana-282 6 17 has have AUX cana-282 6 18 drawn draw VERB cana-282 6 19 on on ADP cana-282 6 20 a a DET cana-282 6 21 graph graph NOUN cana-282 6 22 as as ADP cana-282 6 23 , , PUNCT cana-282 6 24 i= i= PROPN cana-282 6 25 ∫ ∫ PROPN cana-282 6 26 𝑓(π‘₯)𝑑π‘₯ 𝑓(π‘₯)𝑑π‘₯ PROPN cana-282 6 27 𝑏 𝑏 PROPN cana-282 6 28 π‘Ž π‘Ž NOUN cana-282 6 29 is be AUX cana-282 6 30 the the DET cana-282 6 31 sum sum NOUN cana-282 6 32 or or CCONJ cana-282 6 33 total total NOUN cana-282 6 34 of of ADP cana-282 6 35 f(x)dx f(x)dx NOUN cana-282 6 36 on on ADP cana-282 6 37 the the DET cana-282 6 38 interval interval NOUN cana-282 6 39 [ [ X cana-282 6 40 a a X cana-282 6 41 , , PUNCT cana-282 6 42 b b NOUN cana-282 6 43 ] ] X cana-282 6 44 . . PUNCT cana-282 7 1 the the DET cana-282 7 2 value value NOUN cana-282 7 3 of of ADP cana-282 7 4 a a DET cana-282 7 5 definite definite ADJ cana-282 7 6 integral integral NOUN cana-282 7 7 is be AUX cana-282 7 8 calculated calculate VERB cana-282 7 9 via via ADP cana-282 7 10 numerical numerical ADJ cana-282 7 11 integration integration NOUN cana-282 7 12 , , PUNCT cana-282 7 13 which which PRON cana-282 7 14 uses use VERB cana-282 7 15 the the DET cana-282 7 16 integrand integrand NOUN cana-282 7 17 's 's PART cana-282 7 18 approximations approximation NOUN cana-282 7 19 . . PUNCT cana-282 8 1 a a DET cana-282 8 2 quadrature quadrature NOUN cana-282 8 3 function function NOUN cana-282 8 4 is be AUX cana-282 8 5 a a DET cana-282 8 6 way way NOUN cana-282 8 7 to to PART cana-282 8 8 represent represent VERB cana-282 8 9 the the DET cana-282 8 10 area area NOUN cana-282 8 11 under under ADP cana-282 8 12 the the DET cana-282 8 13 curve curve NOUN cana-282 8 14 f(x f(x PROPN cana-282 8 15 ) ) PUNCT cana-282 8 16 , , PUNCT cana-282 8 17 which which PRON cana-282 8 18 is be AUX cana-282 8 19 applied apply VERB cana-282 8 20 to to ADP cana-282 8 21 a a DET cana-282 8 22 single single ADJ cana-282 8 23 variable variable NOUN cana-282 8 24 during during ADP cana-282 8 25 numerical numerical ADJ cana-282 8 26 integration integration NOUN cana-282 8 27 . . PUNCT cana-282 9 1 numerical numerical ADJ cana-282 9 2 integration integration NOUN cana-282 9 3 covers cover VERB cana-282 9 4 all all DET cana-282 9 5 techniques technique NOUN cana-282 9 6 for for ADP cana-282 9 7 estimating estimate VERB cana-282 9 8 the the DET cana-282 9 9 numerical numerical ADJ cana-282 9 10 values value NOUN cana-282 9 11 of of ADP cana-282 9 12 definite definite ADJ cana-282 9 13 integrals integral NOUN cana-282 9 14 , , PUNCT cana-282 9 15 presuming presume VERB cana-282 9 16 there there PRON cana-282 9 17 are be VERB cana-282 9 18 no no DET cana-282 9 19 integrand integrand NOUN cana-282 9 20 singularities singularity NOUN cana-282 9 21 in in ADP cana-282 9 22 the the DET cana-282 9 23 domain domain NOUN cana-282 9 24 . . PUNCT cana-282 10 1 nowadays nowadays ADV cana-282 10 2 , , PUNCT cana-282 10 3 it it PRON cana-282 10 4 is be AUX cana-282 10 5 essential essential ADJ cana-282 10 6 because because SCONJ cana-282 10 7 computers computer NOUN cana-282 10 8 can can AUX cana-282 10 9 connect connect VERB cana-282 10 10 analytical analytical ADJ cana-282 10 11 models model NOUN cana-282 10 12 to to ADP cana-282 10 13 computer computer NOUN cana-282 10 14 processors processor NOUN cana-282 10 15 as as ADV cana-282 10 16 well well ADV cana-282 10 17 as as ADP cana-282 10 18 integrate integrate VERB cana-282 10 19 data datum NOUN cana-282 10 20 in in ADP cana-282 10 21 analytical analytical ADJ cana-282 10 22 ways way NOUN cana-282 10 23 . . PUNCT cana-282 11 1 the the DET cana-282 11 2 concept concept NOUN cana-282 11 3 of of ADP cana-282 11 4 " " PUNCT cana-282 11 5 numerical numerical ADJ cana-282 11 6 integration integration NOUN cana-282 11 7 " " PUNCT cana-282 11 8 was be AUX cana-282 11 9 first first ADV cana-282 11 10 used use VERB cana-282 11 11 in in ADP cana-282 11 12 1915 1915 NUM cana-282 11 13 when when SCONJ cana-282 11 14 david david PROPN cana-282 11 15 gibb gibb PROPN cana-282 11 16 published publish VERB cana-282 11 17 a a DET cana-282 11 18 course course NOUN cana-282 11 19 in in ADP cana-282 11 20 interpolation interpolation NOUN cana-282 11 21 & & CCONJ cana-282 11 22 numeric numeric ADJ cana-282 11 23 integration integration NOUN cana-282 11 24 for for ADP cana-282 11 25 mathematical mathematical ADJ cana-282 11 26 laboratory laboratory NOUN cana-282 11 27 . . PUNCT cana-282 12 1 applied apply VERB cana-282 12 2 mathematics mathematic NOUN cana-282 12 3 , , PUNCT cana-282 12 4 statistics statistic NOUN cana-282 12 5 , , PUNCT cana-282 12 6 economics economic NOUN cana-282 12 7 , , PUNCT cana-282 12 8 and and CCONJ cana-282 12 9 engineering engineering NOUN cana-282 12 10 are be AUX cana-282 12 11 just just ADV cana-282 12 12 a a DET cana-282 12 13 few few ADJ cana-282 12 14 of of ADP cana-282 12 15 the the DET cana-282 12 16 numerous numerous ADJ cana-282 12 17 fields field NOUN cana-282 12 18 where where SCONJ cana-282 12 19 numerical numerical ADJ cana-282 12 20 integration integration NOUN cana-282 12 21 is be AUX cana-282 12 22 used use VERB cana-282 12 23 . . PUNCT cana-282 13 1 numerical numerical ADJ cana-282 13 2 integration integration NOUN cana-282 13 3 can can AUX cana-282 13 4 be be AUX cana-282 13 5 performed perform VERB cana-282 13 6 by by ADP cana-282 13 7 a a DET cana-282 13 8 variety variety NOUN cana-282 13 9 of of ADP cana-282 13 10 techniques technique NOUN cana-282 13 11 , , PUNCT cana-282 13 12 such such ADJ cana-282 13 13 as as ADP cana-282 13 14 euler euler NOUN cana-282 13 15 - - PUNCT cana-282 13 16 maclaurin maclaurin NOUN cana-282 13 17 formula formula NOUN cana-282 13 18 , , PUNCT cana-282 13 19 adaptive adaptive ADJ cana-282 13 20 quadrature quadrature NOUN cana-282 13 21 , , PUNCT cana-282 13 22 gaussian gaussian ADJ cana-282 13 23 integration integration NOUN cana-282 13 24 , , PUNCT cana-282 13 25 & & CCONJ cana-282 13 26 quadrature quadrature NOUN cana-282 13 27 methods method NOUN cana-282 13 28 , , PUNCT cana-282 13 29 which which PRON cana-282 13 30 are be AUX cana-282 13 31 used use VERB cana-282 13 32 to to PART cana-282 13 33 compute compute VERB cana-282 13 34 difficult difficult ADJ cana-282 13 35 - - PUNCT cana-282 13 36 to to ADP cana-282 13 37 - - PUNCT cana-282 13 38 integrate integrate VERB cana-282 13 39 functions function NOUN cana-282 13 40 . . PUNCT cana-282 14 1 several several ADJ cana-282 14 2 authors author NOUN cana-282 14 3 , , PUNCT cana-282 14 4 including include VERB cana-282 14 5 r.l r.l PROPN cana-282 14 6 . . PROPN cana-282 14 7 burden burden NOUN cana-282 14 8 [ [ X cana-282 14 9 2 2 NUM cana-282 14 10 ] ] PUNCT cana-282 14 11 , , PUNCT cana-282 14 12 j.h j.h PROPN cana-282 14 13 . . PROPN cana-282 14 14 mathews mathews PROPN cana-282 14 15 [ [ X cana-282 14 16 3 3 NUM cana-282 14 17 ] ] PUNCT cana-282 14 18 , , PUNCT cana-282 14 19 s.s s.s PROPN cana-282 14 20 . . PROPN cana-282 14 21 sastry sastry NOUN cana-282 15 1 [ [ X cana-282 15 2 1 1 NUM cana-282 15 3 ] ] PUNCT cana-282 15 4 , , PUNCT cana-282 15 5 and and CCONJ cana-282 15 6 others other NOUN cana-282 15 7 , , PUNCT cana-282 15 8 have have AUX cana-282 15 9 written write VERB cana-282 15 10 works work NOUN cana-282 15 11 that that PRON cana-282 15 12 recount recount VERB cana-282 15 13 the the DET cana-282 15 14 various various ADJ cana-282 15 15 formulas formula NOUN cana-282 15 16 for for ADP cana-282 15 17 numerical numerical ADJ cana-282 15 18 integrations integration NOUN cana-282 15 19 . . PUNCT cana-282 16 1 j. j. PROPN cana-282 16 2 oliver oliver PROPN cana-282 17 1 [ [ X cana-282 17 2 4 4 X cana-282 17 3 ] ] PUNCT cana-282 17 4 used use VERB cana-282 17 5 higher high ADJ cana-282 17 6 - - PUNCT cana-282 17 7 order order NOUN cana-282 17 8 formula formula NOUN cana-282 17 9 to to PART cana-282 17 10 investigate investigate VERB cana-282 17 11 the the DET cana-282 17 12 various various ADJ cana-282 17 13 techniques technique NOUN cana-282 17 14 used use VERB cana-282 17 15 to to PART cana-282 17 16 assess assess VERB cana-282 17 17 definite definite ADJ cana-282 17 18 integrals integral NOUN cana-282 17 19 . . PUNCT cana-282 18 1 also also ADV cana-282 18 2 , , PUNCT cana-282 18 3 gerry gerry PROPN cana-282 18 4 sozio sozio PROPN cana-282 19 1 [ [ X cana-282 19 2 5 5 X cana-282 19 3 ] ] PUNCT cana-282 19 4 covered cover VERB cana-282 19 5 a a DET cana-282 19 6 complete complete ADJ cana-282 19 7 description description NOUN cana-282 19 8 of of ADP cana-282 19 9 several several ADJ cana-282 19 10 numerical numerical ADJ cana-282 19 11 integration integration NOUN cana-282 19 12 techniques technique NOUN cana-282 19 13 . . PUNCT cana-282 20 1 numerical numerical ADJ cana-282 20 2 integration integration NOUN cana-282 20 3 is be AUX cana-282 20 4 used use VERB cana-282 20 5 to to PART cana-282 20 6 estimate estimate VERB cana-282 20 7 likelihoods likelihood NOUN cana-282 20 8 and and CCONJ cana-282 20 9 communications communication NOUN cana-282 20 10 on on ADP cana-282 20 11 applied apply VERB cana-282 20 12 nonlinear nonlinear ADJ cana-282 20 13 analysis analysis NOUN cana-282 20 14 issn issn NOUN cana-282 20 15 : : PUNCT cana-282 20 16 1074 1074 NUM cana-282 20 17 - - PUNCT cana-282 20 18 133x 133x NUM cana-282 20 19 vol vol NOUN cana-282 20 20 30 30 NUM cana-282 20 21 no no NOUN cana-282 20 22 . . NOUN cana-282 20 23 4 4 NUM cana-282 20 24 ( ( PUNCT cana-282 20 25 2023 2023 NUM cana-282 20 26 ) ) PUNCT cana-282 20 27 40 40 NUM cana-282 20 28 https://internationalpubls.com https://internationalpubls.com X cana-282 20 29 posterior posterior ADJ cana-282 20 30 distributions distribution NOUN cana-282 20 31 using use VERB cana-282 20 32 bayesian bayesian NOUN cana-282 20 33 techniques technique NOUN cana-282 20 34 [ [ X cana-282 20 35 6 6 NUM cana-282 20 36 ] ] PUNCT cana-282 20 37 . . PUNCT cana-282 21 1 it it PRON cana-282 21 2 also also ADV cana-282 21 3 includes include VERB cana-282 21 4 the the DET cana-282 21 5 value value NOUN cana-282 21 6 of of ADP cana-282 21 7 the the DET cana-282 21 8 definite definite ADJ cana-282 21 9 integral integral ADJ cana-282 21 10 ∫ ∫ PROPN cana-282 21 11 𝑦 𝑦 NOUN cana-282 21 12 𝑑π‘₯ 𝑑π‘₯ VERB cana-282 21 13 𝑏 𝑏 ADP cana-282 21 14 π‘Ž π‘Ž NOUN cana-282 21 15 which which PRON cana-282 21 16 is be AUX cana-282 21 17 calculated calculate VERB cana-282 21 18 by by ADP cana-282 21 19 modifying modify VERB cana-282 21 20 the the DET cana-282 21 21 function function NOUN cana-282 21 22 y y NOUN cana-282 21 23 using use VERB cana-282 21 24 an an DET cana-282 21 25 interpolation interpolation NOUN cana-282 21 26 formula formula NOUN cana-282 21 27 and and CCONJ cana-282 21 28 then then ADV cana-282 21 29 integrating integrate VERB cana-282 21 30 from from ADP cana-282 21 31 a a PRON cana-282 21 32 to to ADP cana-282 21 33 b. b. PROPN cana-282 21 34 we we PRON cana-282 21 35 may may AUX cana-282 21 36 obtain obtain VERB cana-282 21 37 a a DET cana-282 21 38 quadrature quadrature NOUN cana-282 21 39 formula formula NOUN cana-282 21 40 by by ADP cana-282 21 41 this this DET cana-282 21 42 technique technique NOUN cana-282 21 43 with with ADP cana-282 21 44 known know VERB cana-282 21 45 numerical numerical ADJ cana-282 21 46 values value NOUN cana-282 21 47 . . PUNCT cana-282 22 1 it it PRON cana-282 22 2 is be AUX cana-282 22 3 more more ADV cana-282 22 4 essential essential ADJ cana-282 22 5 than than ADP cana-282 22 6 numerical numerical ADJ cana-282 22 7 differentiation differentiation NOUN cana-282 22 8 in in ADP cana-282 22 9 many many ADJ cana-282 22 10 real real ADJ cana-282 22 11 - - PUNCT cana-282 22 12 life life NOUN cana-282 22 13 situations situation NOUN cana-282 22 14 . . PUNCT cana-282 23 1 one one NUM cana-282 23 2 of of ADP cana-282 23 3 the the DET cana-282 23 4 branches branch NOUN cana-282 23 5 connecting connect VERB cana-282 23 6 analytical analytical ADJ cana-282 23 7 calculations calculation NOUN cana-282 23 8 with with ADP cana-282 23 9 computer computer NOUN cana-282 23 10 analysis analysis NOUN cana-282 23 11 is be AUX cana-282 23 12 numerical numerical ADJ cana-282 23 13 integration integration NOUN cana-282 23 14 , , PUNCT cana-282 23 15 and and CCONJ cana-282 23 16 it it PRON cana-282 23 17 plays play VERB cana-282 23 18 a a DET cana-282 23 19 significantly significantly ADV cana-282 23 20 important important ADJ cana-282 23 21 role role NOUN cana-282 23 22 in in ADP cana-282 23 23 mathematics mathematics NOUN cana-282 23 24 affiliate affiliate NOUN cana-282 23 25 today today NOUN cana-282 23 26 . . PUNCT cana-282 24 1 on on ADP cana-282 24 2 the the DET cana-282 24 3 other other ADJ cana-282 24 4 hand hand NOUN cana-282 24 5 , , PUNCT cana-282 24 6 certain certain ADJ cana-282 24 7 researchers researcher NOUN cana-282 24 8 have have AUX cana-282 24 9 previously previously ADV cana-282 24 10 finished finish VERB cana-282 24 11 significant significant ADJ cana-282 24 12 research research NOUN cana-282 24 13 projects project NOUN cana-282 24 14 with with ADP cana-282 24 15 a a DET cana-282 24 16 view view NOUN cana-282 24 17 to to ADP cana-282 24 18 modelling model VERB cana-282 24 19 and and CCONJ cana-282 24 20 developing develop VERB cana-282 24 21 the the DET cana-282 24 22 numerous numerous ADJ cana-282 24 23 domains domain NOUN cana-282 24 24 of of ADP cana-282 24 25 numerical numerical ADJ cana-282 24 26 integration integration NOUN cana-282 24 27 for for ADP cana-282 24 28 multiple multiple ADJ cana-282 24 29 objectives objective NOUN cana-282 24 30 . . PUNCT cana-282 25 1 the the DET cana-282 25 2 analytical analytical ADJ cana-282 25 3 formula formula NOUN cana-282 25 4 for for ADP cana-282 25 5 the the DET cana-282 25 6 kramers kramer NOUN cana-282 25 7 - - PUNCT cana-282 25 8 kronig kronig ADJ cana-282 25 9 transformation transformation NOUN cana-282 25 10 of of ADP cana-282 25 11 a a DET cana-282 25 12 lorentzian lorentzian ADJ cana-282 25 13 function function NOUN cana-282 25 14 was be AUX cana-282 25 15 used use VERB cana-282 25 16 by by ADP cana-282 25 17 ohta ohta PROPN cana-282 25 18 et et PROPN cana-282 25 19 al al PROPN cana-282 25 20 . . PROPN cana-282 25 21 to to PART cana-282 25 22 assess assess VERB cana-282 25 23 several several ADJ cana-282 25 24 numerical numerical ADJ cana-282 25 25 integration integration NOUN cana-282 25 26 methods method NOUN cana-282 25 27 and and CCONJ cana-282 25 28 identify identify VERB cana-282 25 29 the the DET cana-282 25 30 most most ADV cana-282 25 31 efficient efficient ADJ cana-282 25 32 approach approach NOUN cana-282 25 33 [ [ X cana-282 25 34 7 7 NUM cana-282 25 35 ] ] PUNCT cana-282 25 36 . . PUNCT cana-282 26 1 the the DET cana-282 26 2 applications application NOUN cana-282 26 3 of of ADP cana-282 26 4 the the DET cana-282 26 5 maclaurin maclaurin NOUN cana-282 26 6 's 's PART cana-282 26 7 formula formula NOUN cana-282 26 8 , , PUNCT cana-282 26 9 the the DET cana-282 26 10 trapezium trapezium NOUN cana-282 26 11 formula formula NOUN cana-282 26 12 , , PUNCT cana-282 26 13 the the DET cana-282 26 14 simpson simpson NOUN cana-282 26 15 's 's PART cana-282 26 16 formula formula NOUN cana-282 26 17 , , PUNCT cana-282 26 18 and and CCONJ cana-282 26 19 consecutive consecutive ADJ cana-282 26 20 double double ADJ cana-282 26 21 fourier fourier NOUN cana-282 26 22 transform transform NOUN cana-282 26 23 methods method NOUN cana-282 26 24 were be AUX cana-282 26 25 also also ADV cana-282 26 26 contrasted contrast VERB cana-282 26 27 . . PUNCT cana-282 27 1 the the DET cana-282 27 2 electromagnetic electromagnetic ADJ cana-282 27 3 fundamental fundamental ADJ cana-282 27 4 relation relation NOUN cana-282 27 5 's 's PART cana-282 27 6 convolution convolution NOUN cana-282 27 7 integral integral ADJ cana-282 27 8 was be AUX cana-282 27 9 calculated calculate VERB cana-282 27 10 employing employ VERB cana-282 27 11 a a DET cana-282 27 12 newly newly ADV cana-282 27 13 developed develop VERB cana-282 27 14 one one NUM cana-282 27 15 - - PUNCT cana-282 27 16 time time NOUN cana-282 27 17 - - PUNCT cana-282 27 18 step step NOUN cana-282 27 19 recursion recursion NOUN cana-282 27 20 relation relation NOUN cana-282 27 21 and and CCONJ cana-282 27 22 the the DET cana-282 27 23 trapezoidal trapezoidal ADJ cana-282 27 24 rule rule NOUN cana-282 27 25 of of ADP cana-282 27 26 numerical numerical ADJ cana-282 27 27 integration integration NOUN cana-282 27 28 . . PUNCT cana-282 28 1 also also ADV cana-282 28 2 , , PUNCT cana-282 28 3 they they PRON cana-282 28 4 have have AUX cana-282 28 5 compared compare VERB cana-282 28 6 various various ADJ cana-282 28 7 time time NOUN cana-282 28 8 - - PUNCT cana-282 28 9 domain domain NOUN cana-282 28 10 numerical numerical ADJ cana-282 28 11 modelling modelling NOUN cana-282 28 12 strategies strategy NOUN cana-282 28 13 for for ADP cana-282 28 14 material material ADJ cana-282 28 15 dispersion dispersion NOUN cana-282 28 16 in in ADP cana-282 28 17 their their PRON cana-282 28 18 research research NOUN cana-282 28 19 . . PUNCT cana-282 29 1 pennestr pennestr NOUN cana-282 29 2 et et PROPN cana-282 29 3 al al PROPN cana-282 29 4 . . PROPN cana-282 29 5 , , PUNCT cana-282 29 6 on on ADP cana-282 29 7 the the DET cana-282 29 8 other other ADJ cana-282 29 9 hand hand NOUN cana-282 29 10 , , PUNCT cana-282 29 11 presented present VERB cana-282 29 12 and and CCONJ cana-282 29 13 evaluated evaluate VERB cana-282 29 14 eight eight NUM cana-282 29 15 widely widely ADV cana-282 29 16 used use VERB cana-282 29 17 engineering engineering NOUN cana-282 29 18 friction friction NOUN cana-282 29 19 force force NOUN cana-282 29 20 models model NOUN cana-282 29 21 , , PUNCT cana-282 29 22 honed hone VERB cana-282 29 23 in in ADP cana-282 29 24 on on ADP cana-282 29 25 well well ADV cana-282 29 26 - - PUNCT cana-282 29 27 known know VERB cana-282 29 28 friction friction NOUN cana-282 29 29 models model NOUN cana-282 29 30 , , PUNCT cana-282 29 31 and and CCONJ cana-282 29 32 provided provide VERB cana-282 29 33 a a DET cana-282 29 34 review review NOUN cana-282 29 35 and and CCONJ cana-282 29 36 comparison comparison NOUN cana-282 29 37 based base VERB cana-282 29 38 on on ADP cana-282 29 39 numerical numerical ADJ cana-282 29 40 effectiveness effectiveness NOUN cana-282 29 41 [ [ X cana-282 29 42 8 8 NUM cana-282 29 43 ] ] PUNCT cana-282 29 44 . . PUNCT cana-282 30 1 uilhoorn uilhoorn VERB cana-282 30 2 et et PROPN cana-282 30 3 al al PROPN cana-282 30 4 . . PROPN cana-282 30 5 examined examine VERB cana-282 30 6 stiff stiff ADJ cana-282 30 7 and and CCONJ cana-282 30 8 nonstiff nonstiff NOUN cana-282 30 9 solvers solver NOUN cana-282 30 10 , , PUNCT cana-282 30 11 specifically specifically ADV cana-282 30 12 integrated integrate VERB cana-282 30 13 explicit explicit ADJ cana-282 30 14 runge runge NOUN cana-282 30 15 - - PUNCT cana-282 30 16 kutta kutta NOUN cana-282 30 17 schemes scheme NOUN cana-282 30 18 , , PUNCT cana-282 30 19 to to PART cana-282 30 20 discover discover VERB cana-282 30 21 a a DET cana-282 30 22 quick quick ADJ cana-282 30 23 and and CCONJ cana-282 30 24 reliable reliable ADJ cana-282 30 25 spatial spatial ADJ cana-282 30 26 integration integration NOUN cana-282 30 27 solver solver VERB cana-282 30 28 to to PART cana-282 30 29 gather gather VERB cana-282 30 30 gas gas NOUN cana-282 30 31 flow flow NOUN cana-282 30 32 transients transient NOUN cana-282 30 33 within within ADP cana-282 30 34 the the DET cana-282 30 35 context context NOUN cana-282 30 36 of of ADP cana-282 30 37 particle particle NOUN cana-282 30 38 filtering filtering NOUN cana-282 31 1 [ [ X cana-282 31 2 9 9 NUM cana-282 31 3 ] ] PUNCT cana-282 31 4 . . PUNCT cana-282 32 1 bhonsale bhonsale NOUN cana-282 32 2 et et PROPN cana-282 32 3 al al PROPN cana-282 32 4 . . PROPN cana-282 32 5 looked look VERB cana-282 32 6 at at ADP cana-282 32 7 the the DET cana-282 32 8 outcomes outcome NOUN cana-282 32 9 of of ADP cana-282 32 10 three three NUM cana-282 32 11 distinct distinct ADJ cana-282 32 12 numerical numerical ADJ cana-282 32 13 solution solution NOUN cana-282 32 14 methodologies methodology NOUN cana-282 32 15 for for ADP cana-282 32 16 breaking break VERB cana-282 32 17 population population NOUN cana-282 32 18 balance balance NOUN cana-282 32 19 models model VERB cana-282 32 20 the the DET cana-282 32 21 cell cell NOUN cana-282 32 22 average average ADJ cana-282 32 23 method method NOUN cana-282 32 24 , , PUNCT cana-282 32 25 the the DET cana-282 32 26 moving move VERB cana-282 32 27 pivot pivot NOUN cana-282 32 28 method method NOUN cana-282 32 29 , , PUNCT cana-282 32 30 and and CCONJ cana-282 32 31 the the DET cana-282 32 32 fixed fixed ADJ cana-282 32 33 pivot pivot NOUN cana-282 32 34 method method NOUN cana-282 32 35 [ [ X cana-282 32 36 10 10 NUM cana-282 32 37 ] ] PUNCT cana-282 32 38 . . PUNCT cana-282 33 1 additionally additionally ADV cana-282 33 2 , , PUNCT cana-282 33 3 using use VERB cana-282 33 4 these these DET cana-282 33 5 methods method NOUN cana-282 33 6 , , PUNCT cana-282 33 7 concepcion concepcion PROPN cana-282 33 8 ausin ausin PROPN cana-282 33 9 , , PUNCT cana-282 33 10 m. m. NOUN cana-282 33 11 explored explore VERB cana-282 33 12 more more ADV cana-282 33 13 sophisticated sophisticated ADJ cana-282 33 14 numerical numerical ADJ cana-282 33 15 integration integration NOUN cana-282 33 16 techniques technique NOUN cana-282 33 17 and and CCONJ cana-282 33 18 compared compare VERB cana-282 33 19 several several ADJ cana-282 33 20 numerical numerical ADJ cana-282 33 21 integration integration NOUN cana-282 33 22 providers provider NOUN cana-282 33 23 [ [ X cana-282 33 24 11 11 NUM cana-282 33 25 ] ] PUNCT cana-282 33 26 . . PUNCT cana-282 34 1 an an DET cana-282 34 2 integrable integrable ADJ cana-282 34 3 polynomial polynomial ADJ cana-282 34 4 discarding discard VERB cana-282 34 5 taylor taylor PROPN cana-282 34 6 series series NOUN cana-282 34 7 has have AUX cana-282 34 8 been be AUX cana-282 34 9 estimated estimate VERB cana-282 34 10 by by ADP cana-282 34 11 rajesh rajesh PROPN cana-282 34 12 kumar kumar PROPN cana-282 34 13 sinha sinha PROPN cana-282 34 14 and and CCONJ cana-282 34 15 colleagues colleague NOUN cana-282 35 1 [ [ X cana-282 35 2 12 12 NUM cana-282 35 3 ] ] PUNCT cana-282 35 4 . . PUNCT cana-282 36 1 one one NUM cana-282 36 2 of of ADP cana-282 36 3 the the DET cana-282 36 4 most most ADV cana-282 36 5 essential essential ADJ cana-282 36 6 and and CCONJ cana-282 36 7 fundamental fundamental ADJ cana-282 36 8 mathematical mathematical ADJ cana-282 36 9 concepts concept NOUN cana-282 36 10 is be AUX cana-282 36 11 definite definite ADJ cana-282 36 12 integration integration NOUN cana-282 36 13 . . PUNCT cana-282 37 1 and and CCONJ cana-282 37 2 it it PRON cana-282 37 3 has have VERB cana-282 37 4 numerous numerous ADJ cana-282 37 5 uses use NOUN cana-282 37 6 in in ADP cana-282 37 7 areas area NOUN cana-282 37 8 like like ADP cana-282 37 9 physics physics NOUN cana-282 37 10 and and CCONJ cana-282 37 11 engineering engineering NOUN cana-282 37 12 . . PUNCT cana-282 38 1 we we PRON cana-282 38 2 must must AUX cana-282 38 3 compute compute VERB cana-282 38 4 integrals integral NOUN cana-282 38 5 in in ADP cana-282 38 6 a a DET cana-282 38 7 number number NOUN cana-282 38 8 of of ADP cana-282 38 9 real real ADJ cana-282 38 10 - - PUNCT cana-282 38 11 world world NOUN cana-282 38 12 issues issue NOUN cana-282 38 13 . . PUNCT cana-282 39 1 in in ADP cana-282 39 2 mathematics mathematic NOUN cana-282 39 3 , , PUNCT cana-282 39 4 the the DET cana-282 39 5 term term NOUN cana-282 39 6 " " PUNCT cana-282 39 7 quadrature quadrature NOUN cana-282 39 8 " " PUNCT cana-282 39 9 refers refer VERB cana-282 39 10 to to ADP cana-282 39 11 the the DET cana-282 39 12 area area NOUN cana-282 39 13 within within ADP cana-282 39 14 the the DET cana-282 39 15 region region NOUN cana-282 39 16 of of ADP cana-282 39 17 the the DET cana-282 39 18 curve curve NOUN cana-282 39 19 f f PROPN cana-282 39 20 ( ( PUNCT cana-282 39 21 x x X cana-282 39 22 ) ) PUNCT cana-282 39 23 bounded bound VERB cana-282 39 24 by by ADP cana-282 39 25 the the DET cana-282 39 26 ordinates ordinates PROPN cana-282 39 27 π‘₯0 π‘₯0 NOUN cana-282 39 28 , , PUNCT cana-282 39 29 π‘₯𝑛 π‘₯𝑛 PROPN cana-282 39 30 , , PUNCT cana-282 39 31 and and CCONJ cana-282 39 32 the the DET cana-282 39 33 x x NOUN cana-282 39 34 - - NOUN cana-282 39 35 axis axis ADJ cana-282 39 36 . . PUNCT cana-282 40 1 the the DET cana-282 40 2 term term NOUN cana-282 40 3 " " PUNCT cana-282 40 4 cubature cubature NOUN cana-282 40 5 " " PUNCT cana-282 40 6 is be AUX cana-282 40 7 occasionally occasionally ADV cana-282 40 8 used use VERB cana-282 40 9 to to PART cana-282 40 10 describe describe VERB cana-282 40 11 the the DET cana-282 40 12 numerical numerical ADJ cana-282 40 13 integration integration NOUN cana-282 40 14 of of ADP cana-282 40 15 a a DET cana-282 40 16 multiple multiple ADJ cana-282 40 17 integral integral ADJ cana-282 40 18 . . PUNCT cana-282 41 1 numerical numerical ADJ cana-282 41 2 integration integration NOUN cana-282 41 3 issues issue NOUN cana-282 41 4 have have AUX cana-282 41 5 existed exist VERB cana-282 41 6 at at ADP cana-282 41 7 least least ADJ cana-282 41 8 from from ADP cana-282 41 9 the the DET cana-282 41 10 time time NOUN cana-282 41 11 of of ADP cana-282 41 12 the the DET cana-282 41 13 greeks greeks PROPN cana-282 41 14 , , PUNCT cana-282 41 15 when when SCONJ cana-282 41 16 , , PUNCT cana-282 41 17 for for ADP cana-282 41 18 instance instance NOUN cana-282 41 19 , , PUNCT cana-282 41 20 the the DET cana-282 41 21 area area NOUN cana-282 41 22 of of ADP cana-282 41 23 a a DET cana-282 41 24 circle circle NOUN cana-282 41 25 was be AUX cana-282 41 26 calculated calculate VERB cana-282 41 27 by by ADP cana-282 41 28 progressively progressively ADV cana-282 41 29 increasing increase VERB cana-282 41 30 the the DET cana-282 41 31 number number NOUN cana-282 41 32 of of ADP cana-282 41 33 sides side NOUN cana-282 41 34 of of ADP cana-282 41 35 an an DET cana-282 41 36 inscribed inscribe VERB cana-282 41 37 polygon polygon NOUN cana-282 41 38 . . PUNCT cana-282 42 1 the the DET cana-282 42 2 discovery discovery NOUN cana-282 42 3 of of ADP cana-282 42 4 calculus calculus NOUN cana-282 42 5 sparked spark VERB cana-282 42 6 a a DET cana-282 42 7 new new ADJ cana-282 42 8 branch branch NOUN cana-282 42 9 of of ADP cana-282 42 10 study study NOUN cana-282 42 11 that that PRON cana-282 42 12 eventually eventually ADV cana-282 42 13 produced produce VERB cana-282 42 14 the the DET cana-282 42 15 fundamental fundamental ADJ cana-282 42 16 principles principle NOUN cana-282 42 17 of of ADP cana-282 42 18 numerical numerical ADJ cana-282 42 19 integration integration NOUN cana-282 42 20 in in ADP cana-282 42 21 the the DET cana-282 42 22 17th 17th ADJ cana-282 42 23 century century NOUN cana-282 42 24 . . PUNCT cana-282 43 1 following follow VERB cana-282 43 2 those those DET cana-282 43 3 centuries century NOUN cana-282 43 4 of of ADP cana-282 43 5 development development NOUN cana-282 43 6 , , PUNCT cana-282 43 7 the the DET cana-282 43 8 area area NOUN cana-282 43 9 saw see VERB cana-282 43 10 the the DET cana-282 43 11 introduction introduction NOUN cana-282 43 12 of of ADP cana-282 43 13 computers computer NOUN cana-282 43 14 in in ADP cana-282 43 15 more more ADV cana-282 43 16 recent recent ADJ cana-282 43 17 times time NOUN cana-282 43 18 , , PUNCT cana-282 43 19 which which PRON cana-282 43 20 allowed allow VERB cana-282 43 21 for for ADP cana-282 43 22 the the DET cana-282 43 23 employment employment NOUN cana-282 43 24 of of ADP cana-282 43 25 numerous numerous ADJ cana-282 43 26 old old ADJ cana-282 43 27 and and CCONJ cana-282 43 28 new new ADJ cana-282 43 29 algorithms algorithm NOUN cana-282 43 30 to to PART cana-282 43 31 produce produce VERB cana-282 43 32 speedy speedy ADJ cana-282 43 33 and and CCONJ cana-282 43 34 accurate accurate ADJ cana-282 43 35 results result NOUN cana-282 43 36 . . PUNCT cana-282 44 1 several several ADJ cana-282 44 2 scientists scientist NOUN cana-282 44 3 have have AUX cana-282 44 4 already already ADV cana-282 44 5 conducted conduct VERB cana-282 44 6 a a DET cana-282 44 7 substantial substantial ADJ cana-282 44 8 amount amount NOUN cana-282 44 9 of of ADP cana-282 44 10 study study NOUN cana-282 44 11 in in ADP cana-282 44 12 the the DET cana-282 44 13 area area NOUN cana-282 44 14 of of ADP cana-282 44 15 numerical numerical ADJ cana-282 44 16 integration integration NOUN cana-282 44 17 [ [ X cana-282 44 18 13 13 NUM cana-282 44 19 - - SYM cana-282 44 20 17 17 NUM cana-282 44 21 ] ] PUNCT cana-282 44 22 . . PUNCT cana-282 45 1 the the DET cana-282 45 2 numerical numerical ADJ cana-282 45 3 integration integration NOUN cana-282 45 4 ∫ ∫ PROPN cana-282 45 5 𝑓(π‘₯)𝑑π‘₯ 𝑓(π‘₯)𝑑π‘₯ PROPN cana-282 45 6 𝑏 𝑏 NOUN cana-282 45 7 π‘Ž π‘Ž NOUN cana-282 45 8 represents represent VERB cana-282 45 9 the the DET cana-282 45 10 area area NOUN cana-282 45 11 between between ADP cana-282 45 12 the the DET cana-282 45 13 ordinates ordinate NOUN cana-282 45 14 x x PUNCT cana-282 45 15 = = PUNCT cana-282 45 16 a a PRON cana-282 45 17 and and CCONJ cana-282 45 18 x x SYM cana-282 45 19 = = SYM cana-282 45 20 b b PROPN cana-282 45 21 , , PUNCT cana-282 45 22 the the DET cana-282 45 23 x x NOUN cana-282 45 24 - - NOUN cana-282 45 25 axis axis ADJ cana-282 45 26 , , PUNCT cana-282 45 27 and and CCONJ cana-282 45 28 y y PROPN cana-282 45 29 = = PROPN cana-282 45 30 f(x f(x PROPN cana-282 45 31 ) ) PUNCT cana-282 45 32 . . PUNCT cana-282 46 1 f f PROPN cana-282 46 2 ( ( PUNCT cana-282 46 3 x x X cana-282 46 4 ) ) PUNCT cana-282 46 5 must must AUX cana-282 46 6 be be AUX cana-282 46 7 explicitly explicitly ADV cana-282 46 8 specified specify VERB cana-282 46 9 and and CCONJ cana-282 46 10 must must AUX cana-282 46 11 be be AUX cana-282 46 12 integrable integrable ADJ cana-282 46 13 in in ADP cana-282 46 14 order order NOUN cana-282 46 15 for for SCONJ cana-282 46 16 this this DET cana-282 46 17 integration integration NOUN cana-282 46 18 to to PART cana-282 46 19 be be AUX cana-282 46 20 possible possible ADJ cana-282 46 21 . . PUNCT cana-282 47 1 when when SCONJ cana-282 47 2 plotting plot VERB cana-282 47 3 the the DET cana-282 47 4 ordered order VERB cana-282 47 5 pairs pair NOUN cana-282 47 6 ( ( PUNCT cana-282 47 7 π‘₯𝑖 π‘₯𝑖 PROPN cana-282 47 8 , , PUNCT cana-282 47 9 𝑦𝑖 𝑦𝑖 PROPN cana-282 47 10 ) ) PUNCT cana-282 47 11 , , PUNCT cana-282 47 12 communications communication NOUN cana-282 47 13 on on ADP cana-282 47 14 applied apply VERB cana-282 47 15 nonlinear nonlinear ADJ cana-282 47 16 analysis analysis NOUN cana-282 47 17 issn issn NOUN cana-282 47 18 : : PUNCT cana-282 47 19 1074 1074 NUM cana-282 47 20 - - PUNCT cana-282 47 21 133x 133x NUM cana-282 47 22 vol vol NOUN cana-282 47 23 30 30 NUM cana-282 47 24 no no NOUN cana-282 47 25 . . NOUN cana-282 47 26 4 4 NUM cana-282 47 27 ( ( PUNCT cana-282 47 28 2023 2023 NUM cana-282 47 29 ) ) PUNCT cana-282 47 30 41 41 NUM cana-282 48 1 https://internationalpubls.com https://internationalpubls.com X cana-282 48 2 where where SCONJ cana-282 48 3 i i PRON cana-282 48 4 = = NOUN cana-282 48 5 0 0 NUM cana-282 48 6 , , PUNCT cana-282 48 7 1 1 NUM cana-282 48 8 , , PUNCT cana-282 48 9 2 2 NUM cana-282 48 10 … … SYM cana-282 48 11 n n CCONJ cana-282 48 12 , , PUNCT cana-282 48 13 and and CCONJ cana-282 48 14 a a DET cana-282 48 15 straight straight ADJ cana-282 48 16 line line NOUN cana-282 48 17 connects connect VERB cana-282 48 18 any any DET cana-282 48 19 two two NUM cana-282 48 20 successive successive ADJ cana-282 48 21 points point NOUN cana-282 48 22 . . PUNCT cana-282 49 1 numerical numerical ADJ cana-282 49 2 integration integration NOUN cana-282 49 3 problem problem NOUN cana-282 49 4 can can AUX cana-282 49 5 be be AUX cana-282 49 6 defined define VERB cana-282 49 7 as as SCONJ cana-282 49 8 follows follow VERB cana-282 49 9 : : PUNCT cana-282 49 10 let let VERB cana-282 49 11 y y PROPN cana-282 49 12 = = SYM cana-282 49 13 f(x f(x PROPN cana-282 49 14 ) ) PUNCT cana-282 49 15 be be VERB cana-282 49 16 a a DET cana-282 49 17 given give VERB cana-282 49 18 function function NOUN cana-282 49 19 and and CCONJ cana-282 49 20 ( ( PUNCT cana-282 49 21 π‘₯𝑖 π‘₯𝑖 PROPN cana-282 49 22 , , PUNCT cana-282 49 23 𝑦𝑖 𝑦𝑖 PROPN cana-282 49 24 ) ) PUNCT cana-282 49 25 are be AUX cana-282 49 26 the the DET cana-282 49 27 set set NOUN cana-282 49 28 of of ADP cana-282 49 29 n+1 n+1 PROPN cana-282 49 30 ordered order VERB cana-282 49 31 pairs pair NOUN cana-282 49 32 , , PUNCT cana-282 49 33 i i PRON cana-282 49 34 = = NOUN cana-282 49 35 0,1,2 0,1,2 NUM cana-282 49 36 … … SYM cana-282 49 37 n n CCONJ cana-282 49 38 , , PUNCT cana-282 49 39 where where SCONJ cana-282 49 40 f(x f(x PROPN cana-282 49 41 ) ) PUNCT cana-282 49 42 is be AUX cana-282 49 43 not not PART cana-282 49 44 known know VERB cana-282 49 45 explicitly explicitly ADV cana-282 49 46 , , PUNCT cana-282 49 47 then then ADV cana-282 49 48 it it PRON cana-282 49 49 is be AUX cana-282 49 50 necessary necessary ADJ cana-282 49 51 to to PART cana-282 49 52 determine determine VERB cana-282 49 53 ∫ ∫ PROPN cana-282 49 54 𝑦 𝑦 PROPN cana-282 49 55 𝑑π‘₯ 𝑑π‘₯ VERB cana-282 49 56 π‘₯𝑛 π‘₯𝑛 NOUN cana-282 49 57 π‘₯0 π‘₯0 NOUN cana-282 49 58 . . PUNCT cana-282 50 1 numerous numerous ADJ cana-282 50 2 strategies strategy NOUN cana-282 50 3 are be AUX cana-282 50 4 used use VERB cana-282 50 5 in in ADP cana-282 50 6 numerical numerical ADJ cana-282 50 7 integration integration NOUN cana-282 50 8 to to PART cana-282 50 9 determine determine VERB cana-282 50 10 the the DET cana-282 50 11 numerical numerical ADJ cana-282 50 12 value value NOUN cana-282 50 13 of of ADP cana-282 50 14 a a PRON cana-282 50 15 given give VERB cana-282 50 16 integral integral ADJ cana-282 50 17 . . PUNCT cana-282 50 18 numerical numerical ADJ cana-282 50 19 integration integration NOUN cana-282 50 20 is be AUX cana-282 50 21 most most ADV cana-282 50 22 popular popular ADJ cana-282 50 23 technique technique NOUN cana-282 50 24 for for ADP cana-282 50 25 finding find VERB cana-282 50 26 a a DET cana-282 50 27 solution solution NOUN cana-282 50 28 to to ADP cana-282 50 29 some some DET cana-282 50 30 integration integration NOUN cana-282 50 31 problems problem NOUN cana-282 50 32 since since SCONJ cana-282 50 33 some some DET cana-282 50 34 integration integration NOUN cana-282 50 35 problems problem NOUN cana-282 50 36 can can AUX cana-282 50 37 not not PART cana-282 50 38 be be AUX cana-282 50 39 dealt deal VERB cana-282 50 40 with with ADP cana-282 50 41 analytically analytically ADV cana-282 50 42 . . PUNCT cana-282 51 1 numerous numerous ADJ cana-282 51 2 techniques technique NOUN cana-282 51 3 are be AUX cana-282 51 4 used use VERB cana-282 51 5 to to PART cana-282 51 6 approach approach VERB cana-282 51 7 and and CCONJ cana-282 51 8 resolve resolve VERB cana-282 51 9 numerical numerical ADJ cana-282 51 10 integration integration NOUN cana-282 51 11 problems problem NOUN cana-282 51 12 for for ADP cana-282 51 13 unequal unequal ADJ cana-282 51 14 data datum NOUN cana-282 51 15 spaces space NOUN cana-282 51 16 . . PUNCT cana-282 52 1 the the DET cana-282 52 2 process process NOUN cana-282 52 3 of of ADP cana-282 52 4 approximately approximately ADV cana-282 52 5 computing compute VERB cana-282 52 6 an an DET cana-282 52 7 integral integral ADJ cana-282 52 8 using use VERB cana-282 52 9 a a DET cana-282 52 10 numerical numerical ADJ cana-282 52 11 technique technique NOUN cana-282 52 12 is be AUX cana-282 52 13 referred refer VERB cana-282 52 14 to to ADP cana-282 52 15 as as ADP cana-282 52 16 numerical numerical ADJ cana-282 52 17 integration integration NOUN cana-282 52 18 . . PUNCT cana-282 53 1 in in ADP cana-282 53 2 mathematics mathematic NOUN cana-282 53 3 , , PUNCT cana-282 53 4 idea idea NOUN cana-282 53 5 of of ADP cana-282 53 6 numerical numerical ADJ cana-282 53 7 integration integration NOUN cana-282 53 8 is be AUX cana-282 53 9 crucial crucial ADJ cana-282 53 10 . . PUNCT cana-282 54 1 when when SCONJ cana-282 54 2 definite definite ADJ cana-282 54 3 integrals integral NOUN cana-282 54 4 can can AUX cana-282 54 5 not not PART cana-282 54 6 be be AUX cana-282 54 7 solved solve VERB cana-282 54 8 analytically analytically ADV cana-282 54 9 , , PUNCT cana-282 54 10 engineers engineer NOUN cana-282 54 11 and and CCONJ cana-282 54 12 scientists scientist NOUN cana-282 54 13 generally generally ADV cana-282 54 14 use use VERB cana-282 54 15 numerical numerical ADJ cana-282 54 16 integration integration NOUN cana-282 54 17 to to PART cana-282 54 18 approximate approximate VERB cana-282 54 19 definite definite ADJ cana-282 54 20 integrals integral NOUN cana-282 54 21 . . PUNCT cana-282 55 1 numerical numerical ADJ cana-282 55 2 integrations integration NOUN cana-282 55 3 can can AUX cana-282 55 4 be be AUX cana-282 55 5 used use VERB cana-282 55 6 to to PART cana-282 55 7 resolve resolve VERB cana-282 55 8 boundary boundary ADJ cana-282 55 9 value value NOUN cana-282 55 10 and and CCONJ cana-282 55 11 initial initial ADJ cana-282 55 12 value value NOUN cana-282 55 13 issues issue NOUN cana-282 55 14 using use VERB cana-282 55 15 either either CCONJ cana-282 55 16 ordinary ordinary ADJ cana-282 55 17 or or CCONJ cana-282 55 18 partial partial ADJ cana-282 55 19 differential differential ADJ cana-282 55 20 equations equation NOUN cana-282 55 21 . . PUNCT cana-282 56 1 the the DET cana-282 56 2 establishment establishment NOUN cana-282 56 3 of of ADP cana-282 56 4 broad broad ADJ cana-282 56 5 physics physics NOUN cana-282 56 6 laws law NOUN cana-282 56 7 is be AUX cana-282 56 8 typically typically ADV cana-282 56 9 not not PART cana-282 56 10 possible possible ADJ cana-282 56 11 with with ADP cana-282 56 12 numerical numerical ADJ cana-282 56 13 solutions solution NOUN cana-282 56 14 , , PUNCT cana-282 56 15 and and CCONJ cana-282 56 16 they they PRON cana-282 56 17 frequently frequently ADV cana-282 56 18 fail fail VERB cana-282 56 19 to to PART cana-282 56 20 show show VERB cana-282 56 21 how how SCONJ cana-282 56 22 desired desire VERB cana-282 56 23 variables variable NOUN cana-282 56 24 depend depend VERB cana-282 56 25 on on ADP cana-282 56 26 various various ADJ cana-282 56 27 problem problem NOUN cana-282 56 28 - - PUNCT cana-282 56 29 solving solve VERB cana-282 56 30 parameters parameter NOUN cana-282 56 31 . . PUNCT cana-282 57 1 there there PRON cana-282 57 2 are be VERB cana-282 57 3 numerous numerous ADJ cana-282 57 4 numerical numerical ADJ cana-282 57 5 techniques technique NOUN cana-282 57 6 for for ADP cana-282 57 7 approximating approximate VERB cana-282 57 8 the the DET cana-282 57 9 integral integral ADJ cana-282 57 10 for for ADP cana-282 57 11 unequal unequal ADJ cana-282 57 12 data datum NOUN cana-282 57 13 points point NOUN cana-282 57 14 if if SCONJ cana-282 57 15 a a DET cana-282 57 16 smooth smooth ADJ cana-282 57 17 function function NOUN cana-282 57 18 integrand integrand NOUN cana-282 57 19 inside inside ADP cana-282 57 20 a a DET cana-282 57 21 certain certain ADJ cana-282 57 22 interval interval NOUN cana-282 57 23 . . PUNCT cana-282 58 1 which which DET cana-282 58 2 numerically numerically ADV cana-282 58 3 approach approach NOUN cana-282 58 4 produces produce VERB cana-282 58 5 a a DET cana-282 58 6 more more ADV cana-282 58 7 accurate accurate ADJ cana-282 58 8 answer answer NOUN cana-282 58 9 is be AUX cana-282 58 10 the the DET cana-282 58 11 question question NOUN cana-282 58 12 that that PRON cana-282 58 13 emerges emerge VERB cana-282 58 14 . . PUNCT cana-282 59 1 to to PART cana-282 59 2 select select VERB cana-282 59 3 the the DET cana-282 59 4 optimal optimal ADJ cana-282 59 5 numerical numerical ADJ cana-282 59 6 strategy strategy NOUN cana-282 59 7 , , PUNCT cana-282 59 8 we we PRON cana-282 59 9 apply apply VERB cana-282 59 10 a a DET cana-282 59 11 variety variety NOUN cana-282 59 12 of of ADP cana-282 59 13 numerical numerical ADJ cana-282 59 14 integration integration NOUN cana-282 59 15 formulas formula NOUN cana-282 59 16 for for ADP cana-282 59 17 an an DET cana-282 59 18 unequal unequal ADJ cana-282 59 19 data data NOUN cana-282 59 20 space space NOUN cana-282 59 21 . . PUNCT cana-282 60 1 romesh romesh PROPN cana-282 60 2 kumar kumar PROPN cana-282 60 3 muthumalai muthumalai PROPN cana-282 60 4 intended intend VERB cana-282 60 5 to to PART cana-282 60 6 determine determine VERB cana-282 60 7 the the DET cana-282 60 8 error error NOUN cana-282 60 9 of of ADP cana-282 60 10 numerical numerical ADJ cana-282 60 11 integration integration NOUN cana-282 60 12 and and CCONJ cana-282 60 13 differentiation differentiation NOUN cana-282 60 14 . . PUNCT cana-282 61 1 additionally additionally ADV cana-282 61 2 , , PUNCT cana-282 61 3 he he PRON cana-282 61 4 created create VERB cana-282 61 5 a a DET cana-282 61 6 few few ADJ cana-282 61 7 new new ADJ cana-282 61 8 formulas formula NOUN cana-282 61 9 for for ADP cana-282 61 10 numerical numerical ADJ cana-282 61 11 differentiation differentiation NOUN cana-282 61 12 utilising utilise VERB cana-282 61 13 split split VERB cana-282 61 14 differences difference NOUN cana-282 61 15 , , PUNCT cana-282 61 16 which which PRON cana-282 61 17 are be AUX cana-282 61 18 very very ADV cana-282 61 19 helpful helpful ADJ cana-282 61 20 for for ADP cana-282 61 21 estimating estimate VERB cana-282 61 22 derivatives derivative NOUN cana-282 61 23 if if SCONJ cana-282 61 24 additional additional ADJ cana-282 61 25 information information NOUN cana-282 61 26 about about ADP cana-282 61 27 derivatives derivative NOUN cana-282 61 28 becomes become VERB cana-282 61 29 accessible accessible ADJ cana-282 61 30 . . PUNCT cana-282 62 1 selina selina PROPN cana-282 62 2 parvin parvin PROPN cana-282 62 3 , , PUNCT cana-282 62 4 m.r m.r PROPN cana-282 62 5 . . PROPN cana-282 62 6 hossain hossain PROPN cana-282 62 7 , , PUNCT cana-282 62 8 and and CCONJ cana-282 62 9 md md PROPN cana-282 62 10 . . PROPN cana-282 63 1 mamun mamun PROPN cana-282 63 2 - - PUNCT cana-282 63 3 ur ur PROPN cana-282 63 4 - - PUNCT cana-282 63 5 rashid rashid PROPN cana-282 63 6 khan khan PROPN cana-282 63 7 created create VERB cana-282 63 8 a a DET cana-282 63 9 completely completely ADV cana-282 63 10 novel novel ADJ cana-282 63 11 algorithm algorithm NOUN cana-282 63 12 for for ADP cana-282 63 13 numerical numerical ADJ cana-282 63 14 integration integration NOUN cana-282 63 15 techniques technique NOUN cana-282 63 16 for for ADP cana-282 63 17 unequal unequal ADJ cana-282 63 18 data datum NOUN cana-282 63 19 space space NOUN cana-282 63 20 . . PUNCT cana-282 64 1 rajinder rajinder PROPN cana-282 64 2 thukral thukral PROPN cana-282 64 3 follow follow PROPN cana-282 64 4 simpson simpson PROPN cana-282 64 5 's 's PART cana-282 64 6 approach approach NOUN cana-282 64 7 to to PART cana-282 64 8 effectively effectively ADV cana-282 64 9 solve solve VERB cana-282 64 10 nonlinear nonlinear ADJ cana-282 64 11 equations equation NOUN cana-282 64 12 . . PUNCT cana-282 65 1 to to PART cana-282 65 2 find find VERB cana-282 65 3 the the DET cana-282 65 4 zeros zero NOUN cana-282 65 5 of of ADP cana-282 65 6 nonlinear nonlinear ADJ cana-282 65 7 equations equation NOUN cana-282 65 8 , , PUNCT cana-282 65 9 he he PRON cana-282 65 10 improved improve VERB cana-282 65 11 the the DET cana-282 65 12 conventional conventional ADJ cana-282 65 13 simpson simpson PROPN cana-282 65 14 's 's PART cana-282 65 15 third third ADJ cana-282 65 16 order order NOUN cana-282 65 17 approach approach NOUN cana-282 65 18 and and CCONJ cana-282 65 19 created create VERB cana-282 65 20 a a DET cana-282 65 21 novel novel ADJ cana-282 65 22 formula formula NOUN cana-282 65 23 for for ADP cana-282 65 24 second second ADJ cana-282 65 25 order order NOUN cana-282 65 26 derivative derivative ADJ cana-282 65 27 estimation estimation NOUN cana-282 65 28 . . PUNCT cana-282 66 1 in in ADP cana-282 66 2 order order NOUN cana-282 66 3 to to PART cana-282 66 4 solve solve VERB cana-282 66 5 nonlinear nonlinear ADJ cana-282 66 6 equations equation NOUN cana-282 66 7 with with ADP cana-282 66 8 cubic cubic ADJ cana-282 66 9 convergence convergence NOUN cana-282 66 10 , , PUNCT cana-282 66 11 j j PROPN cana-282 66 12 jayakumar jayakumar PROPN cana-282 66 13 generalised generalise VERB cana-282 66 14 simpson simpson PROPN cana-282 66 15 's 's PART cana-282 66 16 newton newton PROPN cana-282 66 17 approach approach NOUN cana-282 66 18 . . PUNCT cana-282 67 1 a a DET cana-282 67 2 novel novel ADJ cana-282 67 3 class class NOUN cana-282 67 4 of of ADP cana-282 67 5 newton newton PROPN cana-282 67 6 's 's PART cana-282 67 7 method method NOUN cana-282 67 8 is be AUX cana-282 67 9 developed develop VERB cana-282 67 10 to to PART cana-282 67 11 solve solve VERB cana-282 67 12 a a DET cana-282 67 13 single single ADJ cana-282 67 14 nonlinear nonlinear ADJ cana-282 67 15 equation equation NOUN cana-282 67 16 for for ADP cana-282 67 17 that that PRON cana-282 68 1 [ [ X cana-282 68 2 18 18 NUM cana-282 68 3 - - SYM cana-282 68 4 21 21 NUM cana-282 68 5 ] ] PUNCT cana-282 68 6 . . PUNCT cana-282 69 1 for for ADP cana-282 69 2 solving solve VERB cana-282 69 3 nonlinear nonlinear ADJ cana-282 69 4 equations equation NOUN cana-282 69 5 , , PUNCT cana-282 69 6 hamidah hamidah PROPN cana-282 69 7 eskandari eskandari PROPN cana-282 69 8 invented invent VERB cana-282 69 9 the the DET cana-282 69 10 simpson simpson NOUN cana-282 69 11 's 's PART cana-282 69 12 algorithm algorithm NOUN cana-282 69 13 . . PUNCT cana-282 70 1 he he PRON cana-282 70 2 suggested suggest VERB cana-282 70 3 using use VERB cana-282 70 4 the the DET cana-282 70 5 integration integration NOUN cana-282 70 6 method method NOUN cana-282 70 7 to to PART cana-282 70 8 pinpoint pinpoint VERB cana-282 70 9 the the DET cana-282 70 10 nonlinear nonlinear ADJ cana-282 70 11 equation equation NOUN cana-282 70 12 's 's PART cana-282 70 13 root root NOUN cana-282 70 14 [ [ X cana-282 70 15 22 22 NUM cana-282 70 16 ] ] PUNCT cana-282 70 17 . . PUNCT cana-282 71 1 it it PRON cana-282 71 2 is be AUX cana-282 71 3 strongly strongly ADV cana-282 71 4 contested contest VERB cana-282 71 5 that that SCONJ cana-282 71 6 the the DET cana-282 71 7 ( ( PUNCT cana-282 71 8 composite composite ADJ cana-282 71 9 ) ) PUNCT cana-282 71 10 trapezoidal trapezoidal ADJ cana-282 71 11 rule rule NOUN cana-282 71 12 , , PUNCT cana-282 71 13 the the DET cana-282 71 14 simplest simple ADJ cana-282 71 15 and and CCONJ cana-282 71 16 likely likely ADV cana-282 71 17 oldest old ADJ cana-282 71 18 algorithm algorithm NOUN cana-282 71 19 for for ADP cana-282 71 20 numerical numerical ADJ cana-282 71 21 quadrature quadrature NOUN cana-282 71 22 , , PUNCT cana-282 71 23 is be AUX cana-282 71 24 adequate adequate ADJ cana-282 71 25 to to PART cana-282 71 26 achieve achieve VERB cana-282 71 27 high high ADJ cana-282 71 28 accuracy accuracy NOUN cana-282 71 29 quadrature quadrature NOUN cana-282 71 30 . . PUNCT cana-282 72 1 this this PRON cana-282 72 2 is be AUX cana-282 72 3 mostly mostly ADV cana-282 72 4 since since SCONJ cana-282 72 5 it it PRON cana-282 72 6 integrates integrate VERB cana-282 72 7 standard standard ADJ cana-282 72 8 integrands integrand NOUN cana-282 72 9 over over ADP cana-282 72 10 a a DET cana-282 72 11 finite finite ADJ cana-282 72 12 interval interval NOUN cana-282 72 13 that that PRON cana-282 72 14 has have VERB cana-282 72 15 a a DET cana-282 72 16 low low ADJ cana-282 72 17 order order NOUN cana-282 72 18 of of ADP cana-282 72 19 exactness exactness NOUN cana-282 72 20 . . PUNCT cana-282 73 1 however however ADV cana-282 73 2 , , PUNCT cana-282 73 3 it it PRON cana-282 73 4 has have AUX cana-282 73 5 been be AUX cana-282 73 6 demonstrated demonstrate VERB cana-282 73 7 that that SCONJ cana-282 73 8 the the DET cana-282 73 9 discretization discretization NOUN cana-282 73 10 error error NOUN cana-282 73 11 of of ADP cana-282 73 12 the the DET cana-282 73 13 trapezoidal trapezoidal ADJ cana-282 73 14 method method NOUN cana-282 73 15 is be AUX cana-282 73 16 exponentially exponentially ADV cana-282 73 17 negligible negligible ADJ cana-282 73 18 in in ADP cana-282 73 19 the the DET cana-282 73 20 reciprocal reciprocal ADJ cana-282 73 21 step step NOUN cana-282 73 22 size size NOUN cana-282 73 23 for for ADP cana-282 73 24 integrals integral NOUN cana-282 73 25 over over ADP cana-282 73 26 the the DET cana-282 73 27 real real ADJ cana-282 73 28 line line NOUN cana-282 73 29 r r NOUN cana-282 73 30 of of ADP cana-282 73 31 analytical analytical ADJ cana-282 73 32 functions function NOUN cana-282 73 33 in in ADP cana-282 73 34 an an DET cana-282 73 35 open open ADJ cana-282 73 36 strip strip NOUN cana-282 73 37 containing contain VERB cana-282 73 38 r. r. PROPN cana-282 73 39 for for ADP cana-282 73 40 numerical numerical ADJ cana-282 73 41 quadrature quadrature NOUN cana-282 73 42 of of ADP cana-282 73 43 analytical analytical ADJ cana-282 73 44 functions function NOUN cana-282 73 45 in in ADP cana-282 73 46 particular particular ADJ cana-282 73 47 situations situation NOUN cana-282 73 48 , , PUNCT cana-282 73 49 the the DET cana-282 73 50 trapezoidal trapezoidal ADJ cana-282 73 51 rule rule NOUN cana-282 73 52 is be AUX cana-282 73 53 thus thus ADV cana-282 73 54 among among ADP cana-282 73 55 the the DET cana-282 73 56 most most ADV cana-282 73 57 effective effective ADJ cana-282 73 58 methods method NOUN cana-282 73 59 . . PUNCT cana-282 74 1 it it PRON cana-282 74 2 is be AUX cana-282 74 3 desirable desirable ADJ cana-282 74 4 to to PART cana-282 74 5 increase increase VERB cana-282 74 6 the the DET cana-282 74 7 decay decay NOUN cana-282 74 8 rate rate NOUN cana-282 74 9 of of ADP cana-282 74 10 the the DET cana-282 74 11 integrand integrand NOUN cana-282 74 12 by by ADP cana-282 74 13 employing employ VERB cana-282 74 14 appropriate appropriate ADJ cana-282 74 15 analytical analytical ADJ cana-282 74 16 transformations transformation NOUN cana-282 74 17 of of ADP cana-282 74 18 the the DET cana-282 74 19 integration integration NOUN cana-282 74 20 variable variable NOUN cana-282 74 21 in in ADP cana-282 74 22 order order NOUN cana-282 74 23 to to PART cana-282 74 24 map map VERB cana-282 74 25 common common ADJ cana-282 74 26 intervals interval NOUN cana-282 74 27 to to ADP cana-282 74 28 the the DET cana-282 74 29 real real ADJ cana-282 74 30 line line NOUN cana-282 74 31 and and CCONJ cana-282 74 32 decrease decrease VERB cana-282 74 33 the the DET cana-282 74 34 number number NOUN cana-282 74 35 of of ADP cana-282 74 36 terms term NOUN cana-282 74 37 in in ADP cana-282 74 38 infinite infinite ADJ cana-282 74 39 trapezoidal trapezoidal ADJ cana-282 74 40 sums sum NOUN cana-282 74 41 . . PUNCT cana-282 75 1 communications communication NOUN cana-282 75 2 on on ADP cana-282 75 3 applied apply VERB cana-282 75 4 nonlinear nonlinear ADJ cana-282 75 5 analysis analysis NOUN cana-282 75 6 issn issn NOUN cana-282 75 7 : : PUNCT cana-282 75 8 1074 1074 NUM cana-282 75 9 - - PUNCT cana-282 75 10 133x 133x NUM cana-282 75 11 vol vol NOUN cana-282 75 12 30 30 NUM cana-282 75 13 no no NOUN cana-282 75 14 . . NOUN cana-282 75 15 4 4 NUM cana-282 75 16 ( ( PUNCT cana-282 75 17 2023 2023 NUM cana-282 75 18 ) ) PUNCT cana-282 75 19 42 42 NUM cana-282 75 20 https://internationalpubls.com https://internationalpubls.com X cana-282 76 1 the the DET cana-282 76 2 modified modify VERB cana-282 76 3 integrand integrand NOUN cana-282 76 4 may may AUX cana-282 76 5 not not PART cana-282 76 6 have have VERB cana-282 76 7 a a DET cana-282 76 8 strip strip NOUN cana-282 76 9 of of ADP cana-282 76 10 analyticality analyticality NOUN cana-282 76 11 since since SCONJ cana-282 76 12 such such ADJ cana-282 76 13 alterations alteration NOUN cana-282 76 14 could could AUX cana-282 76 15 result result VERB cana-282 76 16 in in ADP cana-282 76 17 the the DET cana-282 76 18 emergence emergence NOUN cana-282 76 19 of of ADP cana-282 76 20 new new ADJ cana-282 76 21 singularities singularity NOUN cana-282 76 22 in in ADP cana-282 76 23 the the DET cana-282 76 24 complex complex ADJ cana-282 76 25 plane plane NOUN cana-282 76 26 . . PUNCT cana-282 77 1 goodwin goodwin PROPN cana-282 77 2 in in ADP cana-282 77 3 1949 1949 NUM cana-282 77 4 , , PUNCT cana-282 77 5 schwartz schwartz NOUN cana-282 77 6 in in ADP cana-282 77 7 1969 1969 NUM cana-282 77 8 , , PUNCT cana-282 77 9 and and CCONJ cana-282 77 10 stenger stenger PROPN cana-282 77 11 later later ADV cana-282 77 12 examined examine VERB cana-282 77 13 the the DET cana-282 77 14 trapezoidal trapezoidal ADJ cana-282 77 15 rule rule NOUN cana-282 77 16 in in ADP cana-282 77 17 reference reference NOUN cana-282 77 18 to to ADP cana-282 77 19 the the DET cana-282 77 20 quadrature quadrature NOUN cana-282 77 21 of of ADP cana-282 77 22 analytical analytical ADJ cana-282 77 23 functions function NOUN cana-282 77 24 . . PUNCT cana-282 78 1 takahasi takahasi PROPN cana-282 78 2 and and CCONJ cana-282 78 3 mori mori NOUN cana-282 78 4 in in ADP cana-282 78 5 1973 1973 NUM cana-282 78 6 discussed discuss VERB cana-282 78 7 the the DET cana-282 78 8 possibility possibility NOUN cana-282 78 9 of of ADP cana-282 78 10 adopting adopt VERB cana-282 78 11 transformations transformation NOUN cana-282 78 12 of of ADP cana-282 78 13 the the DET cana-282 78 14 integration integration NOUN cana-282 78 15 variable variable NOUN cana-282 78 16 . . PUNCT cana-282 79 1 although although SCONJ cana-282 79 2 summation summation NOUN cana-282 79 3 and and CCONJ cana-282 79 4 mathematical mathematical ADJ cana-282 79 5 integration integration NOUN cana-282 79 6 are be AUX cana-282 79 7 closely closely ADV cana-282 79 8 connected connect VERB cana-282 79 9 , , PUNCT cana-282 79 10 symbol symbol NOUN cana-282 79 11 of of ADP cana-282 79 12 integration integration NOUN cana-282 79 13 is be AUX cana-282 79 14 a a DET cana-282 79 15 stylized stylize VERB cana-282 79 16 capital capital NOUN cana-282 79 17 " " PUNCT cana-282 79 18 s s X cana-282 79 19 " " PUNCT cana-282 79 20 to to PART cana-282 79 21 emphasize emphasize VERB cana-282 79 22 such such ADJ cana-282 79 23 connection connection NOUN cana-282 79 24 [ [ X cana-282 79 25 23 23 NUM cana-282 79 26 ] ] PUNCT cana-282 79 27 . . PUNCT cana-282 80 1 the the DET cana-282 80 2 development development NOUN cana-282 80 3 of of ADP cana-282 80 4 numerical numerical ADJ cana-282 80 5 methods method NOUN cana-282 80 6 , , PUNCT cana-282 80 7 such such ADJ cana-282 80 8 as as ADP cana-282 80 9 integration integration NOUN cana-282 80 10 , , PUNCT cana-282 80 11 for for ADP cana-282 80 12 numerical numerical ADJ cana-282 80 13 data datum NOUN cana-282 80 14 or or CCONJ cana-282 80 15 advanced advanced ADJ cana-282 80 16 analytics analytic NOUN cana-282 80 17 solutions solution NOUN cana-282 80 18 was be AUX cana-282 80 19 carried carry VERB cana-282 80 20 out out ADP cana-282 80 21 to to PART cana-282 80 22 solve solve VERB cana-282 80 23 various various ADJ cana-282 80 24 real real ADJ cana-282 80 25 - - PUNCT cana-282 80 26 world world NOUN cana-282 80 27 engineering engineering NOUN cana-282 80 28 challenges challenge NOUN cana-282 80 29 . . PUNCT cana-282 81 1 many many ADJ cana-282 81 2 numerical numerical ADJ cana-282 81 3 approaches approach NOUN cana-282 81 4 have have AUX cana-282 81 5 been be AUX cana-282 81 6 developed develop VERB cana-282 81 7 to to PART cana-282 81 8 simplify simplify VERB cana-282 81 9 the the DET cana-282 81 10 process process NOUN cana-282 81 11 to to PART cana-282 81 12 overcome overcome VERB cana-282 81 13 these these DET cana-282 81 14 critical critical ADJ cana-282 81 15 challenges challenge NOUN cana-282 81 16 . . PUNCT cana-282 82 1 to to PART cana-282 82 2 make make VERB cana-282 82 3 life life NOUN cana-282 82 4 easier easy ADJ cana-282 82 5 when when SCONJ cana-282 82 6 facing face VERB cana-282 82 7 these these DET cana-282 82 8 challenging challenging ADJ cana-282 82 9 situations situation NOUN cana-282 82 10 , , PUNCT cana-282 82 11 numerous numerous ADJ cana-282 82 12 numerical numerical ADJ cana-282 82 13 techniques technique NOUN cana-282 82 14 have have AUX cana-282 82 15 been be AUX cana-282 82 16 created create VERB cana-282 82 17 . . PUNCT cana-282 83 1 the the DET cana-282 83 2 trapezoidal trapezoidal ADJ cana-282 83 3 technique technique NOUN cana-282 83 4 , , PUNCT cana-282 83 5 which which PRON cana-282 83 6 is be AUX cana-282 83 7 an an DET cana-282 83 8 enormously enormously ADV cana-282 83 9 popular popular ADJ cana-282 83 10 & & CCONJ cana-282 83 11 easy easy ADJ cana-282 83 12 technique technique NOUN cana-282 83 13 because because SCONJ cana-282 83 14 of of ADP cana-282 83 15 its its PRON cana-282 83 16 relatively relatively ADV cana-282 83 17 high high ADJ cana-282 83 18 accuracy accuracy NOUN cana-282 83 19 , , PUNCT cana-282 83 20 is be AUX cana-282 83 21 one one NUM cana-282 83 22 of of ADP cana-282 83 23 the the DET cana-282 83 24 most most ADV cana-282 83 25 prominent prominent ADJ cana-282 83 26 methods method NOUN cana-282 83 27 for for ADP cana-282 83 28 integration integration NOUN cana-282 83 29 . . PUNCT cana-282 84 1 unfortunately unfortunately ADV cana-282 84 2 , , PUNCT cana-282 84 3 this this DET cana-282 84 4 approach approach NOUN cana-282 84 5 can can AUX cana-282 84 6 not not PART cana-282 84 7 be be AUX cana-282 84 8 turned turn VERB cana-282 84 9 error error NOUN cana-282 84 10 - - PUNCT cana-282 84 11 free free ADJ cana-282 84 12 without without ADP cana-282 84 13 splitting split VERB cana-282 84 14 the the DET cana-282 84 15 integration integration NOUN cana-282 84 16 interval interval NOUN cana-282 84 17 into into ADP cana-282 84 18 numerous numerous ADJ cana-282 84 19 smaller small ADJ cana-282 84 20 intervals interval NOUN cana-282 84 21 , , PUNCT cana-282 84 22 which which PRON cana-282 84 23 may may AUX cana-282 84 24 not not PART cana-282 84 25 always always ADV cana-282 84 26 be be AUX cana-282 84 27 practical practical ADJ cana-282 84 28 . . PUNCT cana-282 85 1 when when SCONJ cana-282 85 2 this this PRON cana-282 85 3 is be AUX cana-282 85 4 n't not PART cana-282 85 5 performed perform VERB cana-282 85 6 , , PUNCT cana-282 85 7 the the DET cana-282 85 8 result result NOUN cana-282 85 9 will will AUX cana-282 85 10 be be AUX cana-282 85 11 inaccurate inaccurate ADJ cana-282 85 12 and and CCONJ cana-282 85 13 will will AUX cana-282 85 14 not not PART cana-282 85 15 be be AUX cana-282 85 16 close close ADJ cana-282 85 17 to to ADP cana-282 85 18 the the DET cana-282 85 19 correct correct ADJ cana-282 85 20 response response NOUN cana-282 85 21 . . PUNCT cana-282 86 1 a a DET cana-282 86 2 improved improve VERB cana-282 86 3 numerical numerical ADJ cana-282 86 4 strategy strategy NOUN cana-282 86 5 based base VERB cana-282 86 6 on on ADP cana-282 86 7 the the DET cana-282 86 8 trapezoidal trapezoidal ADJ cana-282 86 9 rule rule NOUN cana-282 86 10 idea idea NOUN cana-282 86 11 is be AUX cana-282 86 12 suggested suggest VERB cana-282 86 13 in in ADP cana-282 86 14 order order NOUN cana-282 86 15 to to PART cana-282 86 16 obtain obtain VERB cana-282 86 17 findings finding NOUN cana-282 86 18 that that PRON cana-282 86 19 are be AUX cana-282 86 20 substantially substantially ADV cana-282 86 21 more more ADV cana-282 86 22 accurate accurate ADJ cana-282 86 23 even even ADV cana-282 86 24 with with ADP cana-282 86 25 just just ADV cana-282 86 26 one one NUM cana-282 86 27 interval interval NOUN cana-282 86 28 period period NOUN cana-282 86 29 . . PUNCT cana-282 87 1 it it PRON cana-282 87 2 involves involve VERB cana-282 87 3 some some DET cana-282 87 4 additional additional ADJ cana-282 87 5 complexity complexity NOUN cana-282 87 6 but but CCONJ cana-282 87 7 produces produce VERB cana-282 87 8 results result NOUN cana-282 87 9 that that PRON cana-282 87 10 are be AUX cana-282 87 11 significantly significantly ADV cana-282 87 12 more more ADV cana-282 87 13 precise precise ADJ cana-282 87 14 . . PUNCT cana-282 88 1 for for ADP cana-282 88 2 the the DET cana-282 88 3 purpose purpose NOUN cana-282 88 4 of of ADP cana-282 88 5 solving solve VERB cana-282 88 6 engineering engineering NOUN cana-282 88 7 challenges challenge NOUN cana-282 88 8 , , PUNCT cana-282 88 9 numerical numerical ADJ cana-282 88 10 methods method NOUN cana-282 88 11 have have AUX cana-282 88 12 drawn draw VERB cana-282 88 13 the the DET cana-282 88 14 attention attention NOUN cana-282 88 15 of of ADP cana-282 88 16 the the DET cana-282 88 17 research research NOUN cana-282 88 18 community community NOUN cana-282 88 19 . . PUNCT cana-282 89 1 its its PRON cana-282 89 2 effectiveness effectiveness NOUN cana-282 89 3 and and CCONJ cana-282 89 4 the the DET cana-282 89 5 development development NOUN cana-282 89 6 of of ADP cana-282 89 7 high high ADJ cana-282 89 8 - - PUNCT cana-282 89 9 speed speed NOUN cana-282 89 10 calculations calculation NOUN cana-282 89 11 accomplished accomplish VERB cana-282 89 12 on on ADP cana-282 89 13 twenty twenty NUM cana-282 89 14 - - PUNCT cana-282 89 15 first first ADJ cana-282 89 16 - - PUNCT cana-282 89 17 century century NOUN cana-282 89 18 processors processor NOUN cana-282 89 19 are be AUX cana-282 89 20 what what PRON cana-282 89 21 have have AUX cana-282 89 22 sparked spark VERB cana-282 89 23 this this DET cana-282 89 24 interest interest NOUN cana-282 89 25 . . PUNCT cana-282 90 1 a a DET cana-282 90 2 case case NOUN cana-282 90 3 study study NOUN cana-282 90 4 of of ADP cana-282 90 5 the the DET cana-282 90 6 growing grow VERB cana-282 90 7 significance significance NOUN cana-282 90 8 of of ADP cana-282 90 9 numerical numerical ADJ cana-282 90 10 approach approach NOUN cana-282 90 11 capabilities capability NOUN cana-282 90 12 in in ADP cana-282 90 13 engineering engineering NOUN cana-282 90 14 software software NOUN cana-282 90 15 is be AUX cana-282 90 16 matlab matlab PROPN cana-282 90 17 . . PUNCT cana-282 91 1 this this DET cana-282 91 2 book book NOUN cana-282 91 3 is be AUX cana-282 91 4 one one NUM cana-282 91 5 of of ADP cana-282 91 6 the the DET cana-282 91 7 frequently frequently ADV cana-282 91 8 used use VERB cana-282 91 9 textbooks textbook NOUN cana-282 91 10 that that PRON cana-282 91 11 institutions institution NOUN cana-282 91 12 prefer prefer VERB cana-282 91 13 to to PART cana-282 91 14 utilise utilise VERB cana-282 91 15 when when SCONJ cana-282 91 16 teaching teach VERB cana-282 91 17 numerical numerical ADJ cana-282 91 18 techniques technique NOUN cana-282 91 19 to to ADP cana-282 91 20 students student NOUN cana-282 91 21 . . PUNCT cana-282 92 1 as as ADP cana-282 92 2 part part NOUN cana-282 92 3 of of ADP cana-282 92 4 their their PRON cana-282 92 5 higher high ADJ cana-282 92 6 education education NOUN cana-282 92 7 requirements requirement NOUN cana-282 92 8 , , PUNCT cana-282 92 9 all all DET cana-282 92 10 engineers engineer NOUN cana-282 92 11 must must AUX cana-282 92 12 examine examine VERB cana-282 92 13 numerical numerical ADJ cana-282 92 14 methods method NOUN cana-282 92 15 [ [ X cana-282 92 16 23 23 NUM cana-282 92 17 ] ] PUNCT cana-282 92 18 . . PUNCT cana-282 93 1 it it PRON cana-282 93 2 is be AUX cana-282 93 3 recognised recognise VERB cana-282 93 4 as as ADP cana-282 93 5 a a DET cana-282 93 6 great great ADJ cana-282 93 7 resource resource NOUN cana-282 93 8 for for ADP cana-282 93 9 engineers engineer NOUN cana-282 93 10 and and CCONJ cana-282 93 11 students student NOUN cana-282 93 12 to to PART cana-282 93 13 study study VERB cana-282 93 14 , , PUNCT cana-282 93 15 perform perform VERB cana-282 93 16 , , PUNCT cana-282 93 17 and and CCONJ cana-282 93 18 engage engage VERB cana-282 93 19 with with ADP cana-282 93 20 engineering engineering NOUN cana-282 93 21 difficulties difficulty NOUN cana-282 93 22 . . PUNCT cana-282 94 1 it it PRON cana-282 94 2 includes include VERB cana-282 94 3 a a DET cana-282 94 4 wide wide ADJ cana-282 94 5 range range NOUN cana-282 94 6 of of ADP cana-282 94 7 integration integration NOUN cana-282 94 8 techniques technique NOUN cana-282 94 9 , , PUNCT cana-282 94 10 including include VERB cana-282 94 11 the the DET cana-282 94 12 trapezoidal trapezoidal ADJ cana-282 94 13 rule rule NOUN cana-282 94 14 , , PUNCT cana-282 94 15 simpson simpson PROPN cana-282 94 16 rules rule NOUN cana-282 94 17 , , PUNCT cana-282 94 18 romberg romberg PROPN cana-282 94 19 , , PUNCT cana-282 94 20 etc etc X cana-282 94 21 . . X cana-282 94 22 for for ADP cana-282 94 23 the the DET cana-282 94 24 purpose purpose NOUN cana-282 94 25 of of ADP cana-282 94 26 comparison comparison NOUN cana-282 94 27 , , PUNCT cana-282 94 28 this this DET cana-282 94 29 book book NOUN cana-282 94 30 is be AUX cana-282 94 31 used use VERB cana-282 94 32 while while SCONJ cana-282 94 33 discussing discuss VERB cana-282 94 34 the the DET cana-282 94 35 equations equation NOUN cana-282 94 36 and and CCONJ cana-282 94 37 trapezoidal trapezoidal ADJ cana-282 94 38 rule rule NOUN cana-282 94 39 illustrations illustration NOUN cana-282 94 40 . . PUNCT cana-282 95 1 in in ADP cana-282 95 2 [ [ X cana-282 95 3 24 24 NUM cana-282 95 4 ] ] PUNCT cana-282 95 5 , , PUNCT cana-282 95 6 the the DET cana-282 95 7 authors author NOUN cana-282 95 8 tried try VERB cana-282 95 9 to to PART cana-282 95 10 find find VERB cana-282 95 11 a a DET cana-282 95 12 solution solution NOUN cana-282 95 13 to to ADP cana-282 95 14 the the DET cana-282 95 15 runge runge NOUN cana-282 95 16 phenomenon phenomenon NOUN cana-282 95 17 in in ADP cana-282 95 18 order order NOUN cana-282 95 19 to to PART cana-282 95 20 obtain obtain VERB cana-282 95 21 perfection perfection NOUN cana-282 95 22 of of ADP cana-282 95 23 a a DET cana-282 95 24 higher high ADJ cana-282 95 25 order order NOUN cana-282 95 26 compared compare VERB cana-282 95 27 to to ADP cana-282 95 28 that that PRON cana-282 95 29 of of ADP cana-282 95 30 the the DET cana-282 95 31 usual usual ADJ cana-282 95 32 quadrature quadrature NOUN cana-282 95 33 methods method NOUN cana-282 95 34 , , PUNCT cana-282 95 35 which which PRON cana-282 95 36 are be AUX cana-282 95 37 only only ADV cana-282 95 38 capable capable ADJ cana-282 95 39 of of ADP cana-282 95 40 achieving achieve VERB cana-282 95 41 accuracy accuracy NOUN cana-282 95 42 of of ADP cana-282 95 43 the the DET cana-282 95 44 fourth fourth ADJ cana-282 95 45 order order NOUN cana-282 95 46 . . PUNCT cana-282 96 1 utilising utilise VERB cana-282 96 2 the the DET cana-282 96 3 trapezoidal trapezoidal ADJ cana-282 96 4 rule rule NOUN cana-282 96 5 frequency frequency NOUN cana-282 96 6 accuracy accuracy NOUN cana-282 96 7 over over ADP cana-282 96 8 the the DET cana-282 96 9 interior interior NOUN cana-282 96 10 of of ADP cana-282 96 11 the the DET cana-282 96 12 intervals interval NOUN cana-282 96 13 , , PUNCT cana-282 96 14 they they PRON cana-282 96 15 provide provide VERB cana-282 96 16 method method NOUN cana-282 96 17 . . PUNCT cana-282 97 1 the the DET cana-282 97 2 trapezoidal trapezoidal ADJ cana-282 97 3 rule rule NOUN cana-282 97 4 has have AUX cana-282 97 5 also also ADV cana-282 97 6 been be AUX cana-282 97 7 critically critically ADV cana-282 97 8 examined examine VERB cana-282 97 9 from from ADP cana-282 97 10 both both CCONJ cana-282 97 11 a a DET cana-282 97 12 mathematical mathematical ADJ cana-282 97 13 and and CCONJ cana-282 97 14 historical historical ADJ cana-282 97 15 perspective perspective NOUN cana-282 97 16 by by ADP cana-282 97 17 trefethen trefethen NOUN cana-282 97 18 and and CCONJ cana-282 97 19 weideman weideman NOUN cana-282 97 20 , , PUNCT cana-282 97 21 who who PRON cana-282 97 22 also also ADV cana-282 97 23 demonstrated demonstrate VERB cana-282 97 24 how how SCONJ cana-282 97 25 it it PRON cana-282 97 26 is be AUX cana-282 97 27 connected connect VERB cana-282 97 28 to to ADP cana-282 97 29 computational computational ADJ cana-282 97 30 techniques technique NOUN cana-282 97 31 used use VERB cana-282 97 32 in in ADP cana-282 97 33 many many ADJ cana-282 97 34 areas area NOUN cana-282 97 35 of of ADP cana-282 97 36 scientific scientific ADJ cana-282 97 37 computing computing NOUN cana-282 97 38 [ [ X cana-282 97 39 25 25 NUM cana-282 97 40 ] ] PUNCT cana-282 97 41 . . PUNCT cana-282 98 1 it it PRON cana-282 98 2 was be AUX cana-282 98 3 found find VERB cana-282 98 4 to to PART cana-282 98 5 be be AUX cana-282 98 6 associated associate VERB cana-282 98 7 with with ADP cana-282 98 8 approaches approach NOUN cana-282 98 9 for for ADP cana-282 98 10 computing compute VERB cana-282 98 11 the the DET cana-282 98 12 eigenvalues eigenvalue NOUN cana-282 98 13 of of ADP cana-282 98 14 matrices matrix NOUN cana-282 98 15 and and CCONJ cana-282 98 16 operators operator NOUN cana-282 98 17 , , PUNCT cana-282 98 18 as as ADV cana-282 98 19 well well ADV cana-282 98 20 as as ADP cana-282 98 21 with with ADP cana-282 98 22 complex complex ADJ cana-282 98 23 ideas idea NOUN cana-282 98 24 , , PUNCT cana-282 98 25 special special ADJ cana-282 98 26 functions function NOUN cana-282 98 27 , , PUNCT cana-282 98 28 integral integral ADJ cana-282 98 29 equations equation NOUN cana-282 98 30 , , PUNCT cana-282 98 31 and and CCONJ cana-282 98 32 rational rational ADJ cana-282 98 33 approximation approximation NOUN cana-282 98 34 . . PUNCT cana-282 99 1 similar similar ADJ cana-282 99 2 research research NOUN cana-282 99 3 was be AUX cana-282 99 4 done do VERB cana-282 99 5 by by ADP cana-282 99 6 the the DET cana-282 99 7 authors author NOUN cana-282 99 8 of of ADP cana-282 99 9 [ [ X cana-282 99 10 26 26 NUM cana-282 99 11 ] ] PUNCT cana-282 99 12 on on ADP cana-282 99 13 techniques technique NOUN cana-282 99 14 for for ADP cana-282 99 15 extremely extremely ADV cana-282 99 16 precise precise ADJ cana-282 99 17 numerical numerical ADJ cana-282 99 18 integration integration NOUN cana-282 99 19 . . PUNCT cana-282 100 1 they they PRON cana-282 100 2 looked look VERB cana-282 100 3 at at ADP cana-282 100 4 numerous numerous ADJ cana-282 100 5 techniques technique NOUN cana-282 100 6 from from ADP cana-282 100 7 1980 1980 NUM cana-282 100 8 to to ADP cana-282 100 9 the the DET cana-282 100 10 present present NOUN cana-282 100 11 and and CCONJ cana-282 100 12 highlighted highlight VERB cana-282 100 13 a a DET cana-282 100 14 few few ADJ cana-282 100 15 that that PRON cana-282 100 16 , , PUNCT cana-282 100 17 when when SCONJ cana-282 100 18 used use VERB cana-282 100 19 in in ADP cana-282 100 20 accordance accordance NOUN cana-282 100 21 with with ADP cana-282 100 22 effective effective ADJ cana-282 100 23 techniques technique NOUN cana-282 100 24 for for ADP cana-282 100 25 integer integer NOUN cana-282 100 26 detection detection NOUN cana-282 100 27 , , PUNCT cana-282 100 28 required require VERB cana-282 100 29 to to PART cana-282 100 30 allow allow VERB cana-282 100 31 for for ADP cana-282 100 32 analytical analytical ADJ cana-282 100 33 interpretation interpretation NOUN cana-282 100 34 of of ADP cana-282 100 35 numerous numerous ADJ cana-282 100 36 communications communication NOUN cana-282 100 37 on on ADP cana-282 100 38 applied apply VERB cana-282 100 39 nonlinear nonlinear ADJ cana-282 100 40 analysis analysis NOUN cana-282 100 41 issn issn NOUN cana-282 100 42 : : PUNCT cana-282 100 43 1074 1074 NUM cana-282 100 44 - - PUNCT cana-282 100 45 133x 133x NUM cana-282 100 46 vol vol NOUN cana-282 100 47 30 30 NUM cana-282 101 1 no no NOUN cana-282 101 2 . . NOUN cana-282 101 3 4 4 NUM cana-282 101 4 ( ( PUNCT cana-282 101 5 2023 2023 NUM cana-282 101 6 ) ) PUNCT cana-282 101 7 43 43 NUM cana-282 101 8 https://internationalpubls.com https://internationalpubls.com X cana-282 101 9 " " PUNCT cana-282 101 10 impossible impossible ADJ cana-282 101 11 " " PUNCT cana-282 101 12 integrands integrand NOUN cana-282 101 13 before before SCONJ cana-282 101 14 these these DET cana-282 101 15 methods method NOUN cana-282 101 16 were be AUX cana-282 101 17 explored explore VERB cana-282 101 18 . . PUNCT cana-282 102 1 as as ADV cana-282 102 2 far far ADV cana-282 102 3 as as SCONJ cana-282 102 4 we we PRON cana-282 102 5 know know VERB cana-282 102 6 , , PUNCT cana-282 102 7 none none NOUN cana-282 102 8 of of ADP cana-282 102 9 authors author NOUN cana-282 102 10 used use VERB cana-282 102 11 the the DET cana-282 102 12 approaches approach NOUN cana-282 102 13 mentioned mention VERB cana-282 102 14 in in ADP cana-282 102 15 the the DET cana-282 102 16 paper paper NOUN cana-282 102 17 . . PUNCT cana-282 103 1 the the DET cana-282 103 2 following following NOUN cana-282 103 3 is be AUX cana-282 103 4 how how SCONJ cana-282 103 5 the the DET cana-282 103 6 paper paper NOUN cana-282 103 7 is be AUX cana-282 103 8 set set VERB cana-282 103 9 up up ADP cana-282 103 10 : : PUNCT cana-282 103 11 the the DET cana-282 103 12 problem problem NOUN cana-282 103 13 formulation formulation NOUN cana-282 103 14 is be AUX cana-282 103 15 presented present VERB cana-282 103 16 in in ADP cana-282 103 17 section section NOUN cana-282 103 18 2 2 NUM cana-282 103 19 , , PUNCT cana-282 103 20 followed follow VERB cana-282 103 21 by by ADP cana-282 103 22 numerical numerical ADJ cana-282 103 23 examples example NOUN cana-282 103 24 in in ADP cana-282 103 25 section section NOUN cana-282 103 26 3 3 NUM cana-282 103 27 , , PUNCT cana-282 103 28 discussion discussion NOUN cana-282 103 29 of of ADP cana-282 103 30 the the DET cana-282 103 31 findings finding NOUN cana-282 103 32 in in ADP cana-282 103 33 section section NOUN cana-282 103 34 4 4 NUM cana-282 103 35 , , PUNCT cana-282 103 36 and and CCONJ cana-282 103 37 the the DET cana-282 103 38 study study NOUN cana-282 103 39 's 's PART cana-282 103 40 conclusion conclusion NOUN cana-282 103 41 in in ADP cana-282 103 42 section section NOUN cana-282 103 43 4 4 NUM cana-282 103 44 . . NOUN cana-282 103 45 2 2 NUM cana-282 103 46 . . NOUN cana-282 103 47 material material NOUN cana-282 103 48 and and CCONJ cana-282 103 49 methods method NOUN cana-282 103 50 the the DET cana-282 103 51 numerical numerical ADJ cana-282 103 52 techniques technique NOUN cana-282 103 53 are be AUX cana-282 103 54 used use VERB cana-282 103 55 to to PART cana-282 103 56 determine determine VERB cana-282 103 57 the the DET cana-282 103 58 actual actual ADJ cana-282 103 59 and and CCONJ cana-282 103 60 approximation approximation NOUN cana-282 103 61 solution solution NOUN cana-282 103 62 of of ADP cana-282 103 63 nonlinear nonlinear ADJ cana-282 103 64 algebraic algebraic ADJ cana-282 103 65 , , PUNCT cana-282 103 66 transcendental transcendental ADJ cana-282 103 67 equation equation NOUN cana-282 103 68 ∫ ∫ PROPN cana-282 103 69 1 1 NUM cana-282 103 70 1+π‘₯2 1+π‘₯2 PROPN cana-282 103 71 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 103 72 , , PUNCT cana-282 103 73 2 2 NUM cana-282 103 74 1 1 NUM cana-282 103 75 ∫ ∫ NOUN cana-282 103 76 1 1 NUM cana-282 104 1 1+π‘₯ 1+π‘₯ PROPN cana-282 104 2 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 104 3 , , PUNCT cana-282 104 4 4 4 NUM cana-282 104 5 2 2 NUM cana-282 104 6 ∫ ∫ NOUN cana-282 104 7 𝑒π‘₯3 𝑒π‘₯3 PROPN cana-282 104 8 1 1 NUM cana-282 104 9 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 104 10 , , PUNCT cana-282 104 11 ∫ ∫ PROPN cana-282 104 12 log log VERB cana-282 104 13 π‘₯ π‘₯ NOUN cana-282 104 14 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 104 15 , , PUNCT cana-282 104 16 5 5 NUM cana-282 104 17 3 3 NUM cana-282 104 18 ∫ ∫ NOUN cana-282 104 19 2π‘₯2 2π‘₯2 NUM cana-282 104 20 1 1 NUM cana-282 104 21 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 104 22 , , PUNCT cana-282 104 23 ∫ ∫ PROPN cana-282 104 24 sec sec PROPN cana-282 105 1 π‘₯ π‘₯ PROPN cana-282 105 2 tan tan NOUN cana-282 105 3 π‘₯ π‘₯ X cana-282 105 4 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 105 5 , , PUNCT cana-282 105 6 1 1 NUM cana-282 105 7 0 0 NUM cana-282 105 8 and and CCONJ cana-282 105 9 ∫ ∫ PROPN cana-282 105 10 𝑒π‘₯1 𝑒π‘₯1 X cana-282 105 11 0 0 PUNCT cana-282 106 1 + + CCONJ cana-282 106 2 sin sin VERB cana-282 106 3 π‘₯ π‘₯ NOUN cana-282 106 4 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 106 5 trapezoidal trapezoidal ADJ cana-282 106 6 method method NOUN cana-282 106 7 and and CCONJ cana-282 106 8 modified modified ADJ cana-282 106 9 trapezoidal trapezoidal ADJ cana-282 106 10 method method NOUN cana-282 106 11 : : PUNCT cana-282 106 12 the the DET cana-282 106 13 technique technique NOUN cana-282 106 14 of of ADP cana-282 106 15 calculating calculate VERB cana-282 106 16 the the DET cana-282 106 17 value value NOUN cana-282 106 18 of of ADP cana-282 106 19 definite definite ADJ cana-282 106 20 integrals integral NOUN cana-282 106 21 is be AUX cana-282 106 22 known know VERB cana-282 106 23 as as ADP cana-282 106 24 numerical numerical ADJ cana-282 106 25 integration integration NOUN cana-282 106 26 in in ADP cana-282 106 27 numerical numerical ADJ cana-282 106 28 analysis analysis NOUN cana-282 106 29 . . PUNCT cana-282 107 1 when when SCONJ cana-282 107 2 an an DET cana-282 107 3 analytical analytical ADJ cana-282 107 4 technique technique NOUN cana-282 107 5 failed fail VERB cana-282 107 6 or or CCONJ cana-282 107 7 when when SCONJ cana-282 107 8 an an DET cana-282 107 9 integrand integrand NOUN cana-282 107 10 function function NOUN cana-282 107 11 was be AUX cana-282 107 12 provided provide VERB cana-282 107 13 in in ADP cana-282 107 14 a a DET cana-282 107 15 data datum NOUN cana-282 107 16 table table NOUN cana-282 107 17 . . PUNCT cana-282 108 1 it it PRON cana-282 108 2 has have VERB cana-282 108 3 several several ADJ cana-282 108 4 uses use NOUN cana-282 108 5 in in ADP cana-282 108 6 the the DET cana-282 108 7 fields field NOUN cana-282 108 8 of of ADP cana-282 108 9 engineering engineering NOUN cana-282 108 10 and and CCONJ cana-282 108 11 physics physics NOUN cana-282 108 12 . . PUNCT cana-282 109 1 in in ADP cana-282 109 2 physics physics PROPN cana-282 109 3 , , PUNCT cana-282 109 4 numerical numerical ADJ cana-282 109 5 integration integration NOUN cana-282 109 6 problems problem NOUN cana-282 109 7 sometimes sometimes ADV cana-282 109 8 require require VERB cana-282 109 9 first first ADV cana-282 109 10 subdividing subdivide VERB cana-282 109 11 the the DET cana-282 109 12 integrand integrand NOUN cana-282 109 13 into into ADP cana-282 109 14 an an DET cana-282 109 15 equidistant equidistant ADJ cana-282 109 16 mesh mesh NOUN cana-282 109 17 and and CCONJ cana-282 109 18 then then ADV cana-282 109 19 applying apply VERB cana-282 109 20 a a DET cana-282 109 21 basic basic ADJ cana-282 109 22 rule rule NOUN cana-282 109 23 of of ADP cana-282 109 24 integration integration NOUN cana-282 109 25 to to PART cana-282 109 26 finally finally ADV cana-282 109 27 attain attain VERB cana-282 109 28 the the DET cana-282 109 29 needed need VERB cana-282 109 30 precision precision NOUN cana-282 109 31 . . PUNCT cana-282 110 1 assume assume VERB cana-282 110 2 that that SCONJ cana-282 110 3 a a DET cana-282 110 4 continuous continuous ADJ cana-282 110 5 function function NOUN cana-282 110 6 y y PROPN cana-282 110 7 = = SYM cana-282 110 8 f(x f(x PROPN cana-282 110 9 ) ) PUNCT cana-282 110 10 defined define VERB cana-282 110 11 on on ADP cana-282 110 12 [ [ X cana-282 110 13 a a DET cana-282 110 14 , , PUNCT cana-282 110 15 b b NOUN cana-282 110 16 ] ] X cana-282 110 17 . . PUNCT cana-282 111 1 the the DET cana-282 111 2 interval interval NOUN cana-282 111 3 [ [ X cana-282 111 4 a a X cana-282 111 5 , , PUNCT cana-282 111 6 b b NOUN cana-282 111 7 ] ] X cana-282 111 8 divided divide VERB cana-282 111 9 into into ADP cana-282 111 10 n n CCONJ cana-282 111 11 equal equal ADJ cana-282 111 12 sub sub NOUN cana-282 111 13 – – PUNCT cana-282 111 14 intervals interval NOUN cana-282 111 15 each each PRON cana-282 111 16 of of ADP cana-282 111 17 width width ADJ cana-282 111 18 h h NOUN cana-282 111 19 = = SYM cana-282 111 20 s.t s.t PROPN cana-282 111 21 . . PROPN cana-282 111 22 a a DET cana-282 111 23 = = X cana-282 111 24 π‘₯0 π‘₯0 PROPN cana-282 111 25 < < X cana-282 111 26 π‘₯1 π‘₯1 NOUN cana-282 111 27 < < X cana-282 111 28 π‘₯2 π‘₯2 PROPN cana-282 111 29 < < X cana-282 111 30 β‹― β‹― X cana-282 111 31 < < X cana-282 111 32 π‘₯𝑛 π‘₯𝑛 PROPN cana-282 111 33 = = SYM cana-282 111 34 𝑏 𝑏 NOUN cana-282 111 35 let let VERB cana-282 111 36 f(x f(x PROPN cana-282 111 37 ) ) PUNCT cana-282 111 38 = = SYM cana-282 111 39 pn(x pn(x X cana-282 111 40 ) ) PUNCT cana-282 111 41 = = PUNCT cana-282 112 1 βˆ‘ βˆ‘ PUNCT cana-282 112 2 πΏπ‘˜(π‘₯)𝑓(π‘₯π‘˜)𝑛 πΏπ‘˜(π‘₯)𝑓(π‘₯π‘˜)𝑛 PROPN cana-282 112 3 π‘˜=0 π‘˜=0 PROPN cana-282 112 4 ... ... PUNCT cana-282 112 5 ( ( PUNCT cana-282 112 6 1 1 X cana-282 112 7 ) ) PUNCT cana-282 112 8 where where SCONJ cana-282 112 9 pn(x pn(x X cana-282 112 10 ) ) PUNCT cana-282 112 11 is be AUX cana-282 112 12 polynomial polynomial ADJ cana-282 112 13 of of ADP cana-282 112 14 degree degree NOUN cana-282 112 15 ≀ ≀ NUM cana-282 112 16 n n CCONJ cana-282 112 17 & & CCONJ cana-282 112 18 lk(x lk(x PROPN cana-282 112 19 ) ) PUNCT cana-282 113 1 = = PRON cana-282 113 2 ( ( PUNCT cana-282 113 3 π‘₯βˆ’π‘₯0)(π‘₯βˆ’π‘₯1) π‘₯βˆ’π‘₯0)(π‘₯βˆ’π‘₯1) NUM cana-282 113 4 … … SYM cana-282 113 5 … … SYM cana-282 113 6 (π‘₯βˆ’π‘₯π‘˜βˆ’1)(π‘₯ (π‘₯βˆ’π‘₯π‘˜βˆ’1)(π‘₯ X cana-282 113 7 βˆ’π‘₯π‘˜+1) βˆ’π‘₯π‘˜+1) PROPN cana-282 113 8 … … PUNCT cana-282 113 9 … … PUNCT cana-282 113 10 .(π‘₯βˆ’π‘₯𝑛 .(π‘₯βˆ’π‘₯𝑛 NUM cana-282 113 11 ) ) PUNCT cana-282 113 12 ( ( PUNCT cana-282 113 13 π‘₯π‘˜βˆ’π‘₯0 π‘₯π‘˜βˆ’π‘₯0 NOUN cana-282 113 14 ) ) PUNCT cana-282 113 15 ( ( PUNCT cana-282 113 16 π‘₯π‘˜βˆ’π‘₯1) π‘₯π‘˜βˆ’π‘₯1) NUM cana-282 113 17 … … PUNCT cana-282 113 18 .. .. PUNCT cana-282 113 19 (π‘₯π‘˜βˆ’π‘₯π‘˜βˆ’1)(π‘₯π‘˜βˆ’π‘₯π‘˜+1) (π‘₯π‘˜βˆ’π‘₯π‘˜βˆ’1)(π‘₯π‘˜βˆ’π‘₯π‘˜+1) X cana-282 113 20 … … PUNCT cana-282 113 21 … … PUNCT cana-282 113 22 (π‘₯π‘˜βˆ’π‘₯𝑛 (π‘₯π‘˜βˆ’π‘₯𝑛 NOUN cana-282 113 23 ) ) PUNCT cana-282 113 24 integrating integrate VERB cana-282 113 25 equation equation NOUN cana-282 113 26 ( ( PUNCT cana-282 113 27 1 1 X cana-282 113 28 ) ) PUNCT cana-282 113 29 w.r.t w.r.t NOUN cana-282 113 30 . . PUNCT cana-282 114 1 x x PUNCT cana-282 115 1 over over ADP cana-282 115 2 limit limit VERB cana-282 115 3 a a DET cana-282 115 4 to to ADP cana-282 115 5 b b PROPN cana-282 115 6 ∫ ∫ PROPN cana-282 115 7 𝑓(π‘₯)𝑑π‘₯ 𝑓(π‘₯)𝑑π‘₯ PROPN cana-282 115 8 𝑏 𝑏 NOUN cana-282 115 9 π‘Ž π‘Ž PROPN cana-282 115 10 = = SYM cana-282 115 11 ∫ ∫ NOUN cana-282 115 12 𝑝𝑛(x)dx 𝑝𝑛(x)dx NOUN cana-282 115 13 𝑏 𝑏 PROPN cana-282 115 14 π‘Ž π‘Ž X cana-282 115 15 = = SYM cana-282 115 16 ∫ ∫ PROPN cana-282 115 17 [ [ PUNCT cana-282 115 18 𝑏 𝑏 PROPN cana-282 115 19 π‘Ž π‘Ž PROPN cana-282 115 20 βˆ‘ βˆ‘ PROPN cana-282 115 21 πΏπ‘˜(π‘₯)𝑓(π‘₯π‘˜)𝑛 πΏπ‘˜(π‘₯)𝑓(π‘₯π‘˜)𝑛 PROPN cana-282 115 22 π‘˜=0 π‘˜=0 X cana-282 115 23 ] ] X cana-282 115 24 𝑑π‘₯ 𝑑π‘₯ SCONJ cana-282 115 25 communications communication NOUN cana-282 115 26 on on ADP cana-282 115 27 applied apply VERB cana-282 115 28 nonlinear nonlinear ADJ cana-282 115 29 analysis analysis NOUN cana-282 115 30 issn issn NOUN cana-282 115 31 : : PUNCT cana-282 115 32 1074 1074 NUM cana-282 115 33 - - PUNCT cana-282 115 34 133x 133x NUM cana-282 115 35 vol vol NOUN cana-282 115 36 30 30 NUM cana-282 115 37 no no NOUN cana-282 115 38 . . NOUN cana-282 115 39 4 4 NUM cana-282 115 40 ( ( PUNCT cana-282 115 41 2023 2023 NUM cana-282 115 42 ) ) PUNCT cana-282 116 1 44 44 NUM cana-282 116 2 https://internationalpubls.com https://internationalpubls.com X cana-282 116 3 ∫ ∫ PROPN cana-282 116 4 𝑓(π‘₯)𝑑π‘₯ 𝑓(π‘₯)𝑑π‘₯ VERB cana-282 116 5 𝑏 𝑏 NOUN cana-282 116 6 π‘Ž π‘Ž PROPN cana-282 116 7 = = SYM cana-282 116 8 ∫ ∫ NOUN cana-282 116 9 𝑝𝑛(x)dx 𝑝𝑛(x)dx NOUN cana-282 116 10 𝑏 𝑏 NOUN cana-282 116 11 π‘Ž π‘Ž NOUN cana-282 116 12 = = SYM cana-282 116 13 βˆ‘ βˆ‘ PROPN cana-282 116 14 [ [ PUNCT cana-282 116 15 ∫ ∫ PROPN cana-282 116 16 πΏπ‘˜(π‘₯)𝑑π‘₯ πΏπ‘˜(π‘₯)𝑑π‘₯ X cana-282 116 17 ] ] X cana-282 116 18 𝑓(π‘₯π‘˜ 𝑓(π‘₯π‘˜ PROPN cana-282 116 19 ) ) PUNCT cana-282 116 20 𝑏 𝑏 PROPN cana-282 116 21 π‘Ž π‘Ž PROPN cana-282 116 22 𝑛 𝑛 PROPN cana-282 116 23 π‘˜=0 π‘˜=0 PRON cana-282 116 24 equation equation NOUN cana-282 116 25 ( ( PUNCT cana-282 116 26 2 2 X cana-282 116 27 ) ) PUNCT cana-282 116 28 is be AUX cana-282 116 29 known know VERB cana-282 116 30 as as ADP cana-282 116 31 newton newton PROPN cana-282 116 32 – – PUNCT cana-282 116 33 cotes cote NOUN cana-282 116 34 formula formula NOUN cana-282 116 35 . . PUNCT cana-282 117 1 put put VERB cana-282 117 2 n n NOUN cana-282 117 3 = = SYM cana-282 117 4 1 1 NUM cana-282 117 5 in in ADP cana-282 117 6 equation equation NOUN cana-282 117 7 ( ( PUNCT cana-282 117 8 2 2 NUM cana-282 117 9 ) ) PUNCT cana-282 117 10 , , PUNCT cana-282 117 11 we we PRON cana-282 117 12 get get VERB cana-282 117 13 ∫ ∫ PROPN cana-282 117 14 𝑓(π‘₯)𝑑π‘₯ 𝑓(π‘₯)𝑑π‘₯ VERB cana-282 117 15 𝑏 𝑏 NOUN cana-282 117 16 π‘Ž π‘Ž NOUN cana-282 117 17 = = SYM cana-282 117 18 ∫ ∫ NOUN cana-282 117 19 𝑝1(x)dx 𝑝1(x)dx VERB cana-282 117 20 𝑏 𝑏 PROPN cana-282 117 21 π‘Ž π‘Ž X cana-282 117 22 = = PUNCT cana-282 117 23 βˆ‘ βˆ‘ PUNCT cana-282 117 24 π΄π‘˜π‘“(π‘₯π‘˜)1 π΄π‘˜π‘“(π‘₯π‘˜)1 PROPN cana-282 117 25 π‘˜=0 π‘˜=0 VERB cana-282 117 26 ∫ ∫ PROPN cana-282 117 27 𝑓(π‘₯)𝑑π‘₯ 𝑓(π‘₯)𝑑π‘₯ VERB cana-282 117 28 𝑏 𝑏 PROPN cana-282 117 29 π‘Ž π‘Ž NOUN cana-282 117 30 = = SYM cana-282 117 31 ∫ ∫ NOUN cana-282 117 32 𝑝1(x)dx 𝑝1(x)dx VERB cana-282 117 33 𝑏 𝑏 PROPN cana-282 117 34 π‘Ž π‘Ž PROPN cana-282 117 35 = = SYM cana-282 117 36 𝐴0f(π‘₯0 𝐴0f(π‘₯0 PROPN cana-282 117 37 ) ) PUNCT cana-282 118 1 + + CCONJ cana-282 118 2 𝐴1f(π‘₯1 𝐴1f(π‘₯1 PROPN cana-282 118 3 ) ) PUNCT cana-282 118 4 ∫ ∫ PROPN cana-282 118 5 𝑓(π‘₯)𝑑π‘₯ 𝑓(π‘₯)𝑑π‘₯ PROPN cana-282 118 6 𝑏 𝑏 PROPN cana-282 118 7 π‘Ž π‘Ž NOUN cana-282 118 8 = = SYM cana-282 118 9 ∫ ∫ NOUN cana-282 118 10 𝑝1(x)dx 𝑝1(x)dx VERB cana-282 118 11 𝑏 𝑏 PRON cana-282 118 12 π‘Ž π‘Ž PROPN cana-282 118 13 = = SYM cana-282 118 14 𝐴0𝑦0 𝐴0𝑦0 NOUN cana-282 118 15 + + NOUN cana-282 118 16 𝐴1𝑦1 𝐴1𝑦1 NOUN cana-282 118 17 … … SYM cana-282 118 18 (3 (3 NUM cana-282 118 19 ) ) PUNCT cana-282 118 20 now now ADV cana-282 118 21 we we PRON cana-282 118 22 find find VERB cana-282 118 23 𝐴0 𝐴0 PROPN cana-282 118 24 & & CCONJ cana-282 118 25 𝐴1 𝐴1 PROPN cana-282 118 26 𝐴0 𝐴0 PROPN cana-282 119 1 = = SYM cana-282 119 2 ∫ ∫ PROPN cana-282 119 3 π‘₯βˆ’π‘ π‘₯βˆ’π‘ NOUN cana-282 119 4 π‘Žβˆ’π‘ π‘Žβˆ’π‘ NOUN cana-282 119 5 𝑏 𝑏 PROPN cana-282 119 6 π‘Ž π‘Ž PRON cana-282 119 7 dx dx PROPN cana-282 119 8 = = SYM cana-282 119 9 ∫ ∫ PROPN cana-282 119 10 π‘₯βˆ’π‘₯1 π‘₯βˆ’π‘₯1 NOUN cana-282 119 11 π‘₯0βˆ’π‘₯1 π‘₯0βˆ’π‘₯1 NUM cana-282 119 12 π‘₯1 π‘₯1 NOUN cana-282 119 13 π‘₯0 π‘₯0 NOUN cana-282 119 14 dx dx PROPN cana-282 119 15 = = SYM cana-282 119 16 ∫ ∫ PROPN cana-282 119 17 π‘₯βˆ’π‘₯0βˆ’β„Ž π‘₯βˆ’π‘₯0βˆ’β„Ž VERB cana-282 119 18 π‘₯0βˆ’π‘₯0βˆ’β„Ž π‘₯0βˆ’π‘₯0βˆ’β„Ž ADV cana-282 119 19 π‘₯0+β„Ž π‘₯0+β„Ž PROPN cana-282 119 20 𝑑0 𝑑0 PROPN cana-282 119 21 dx dx PROPN cana-282 119 22 after after ADP cana-282 119 23 solve solve NOUN cana-282 119 24 above above ADV cana-282 119 25 we we PRON cana-282 119 26 get get VERB cana-282 119 27 𝐴0 𝐴0 NOUN cana-282 119 28 = = PUNCT cana-282 119 29 β„Ž β„Ž X cana-282 119 30 2 2 NUM cana-282 119 31 𝐴1 𝐴1 NOUN cana-282 119 32 = = SYM cana-282 120 1 ∫ ∫ PROPN cana-282 120 2 π‘₯βˆ’π‘Ž π‘₯βˆ’π‘Ž PROPN cana-282 121 1 π‘βˆ’π‘Ž π‘βˆ’π‘Ž VERB cana-282 121 2 𝑏 𝑏 PROPN cana-282 121 3 π‘Ž π‘Ž NOUN cana-282 121 4 dx dx PROPN cana-282 121 5 = = SYM cana-282 121 6 ∫ ∫ PROPN cana-282 122 1 π‘₯βˆ’π‘₯0 π‘₯βˆ’π‘₯0 NUM cana-282 122 2 π‘₯1βˆ’π‘₯0 π‘₯1βˆ’π‘₯0 NUM cana-282 122 3 π‘₯1 π‘₯1 NOUN cana-282 122 4 π‘₯0 π‘₯0 NOUN cana-282 122 5 dx dx PROPN cana-282 122 6 = = SYM cana-282 122 7 ∫ ∫ PROPN cana-282 122 8 π‘₯βˆ’π‘₯0βˆ’β„Ž π‘₯βˆ’π‘₯0βˆ’β„Ž PROPN cana-282 122 9 βˆ’β„Ž βˆ’β„Ž NOUN cana-282 122 10 π‘₯0+β„Ž π‘₯0+β„Ž PROPN cana-282 122 11 𝑑0 𝑑0 PROPN cana-282 122 12 dx dx PROPN cana-282 122 13 after after ADP cana-282 122 14 solving solve VERB cana-282 122 15 we we PRON cana-282 122 16 get get VERB cana-282 122 17 𝐴1 𝐴1 PROPN cana-282 122 18 = = PUNCT cana-282 122 19 β„Ž β„Ž X cana-282 122 20 2 2 NUM cana-282 122 21 put put VERB cana-282 122 22 values value NOUN cana-282 122 23 of of ADP cana-282 122 24 𝐴0 𝐴0 PROPN cana-282 122 25 & & CCONJ cana-282 122 26 𝐴1 𝐴1 PROPN cana-282 122 27 in in ADP cana-282 122 28 ( ( PUNCT cana-282 122 29 3 3 NUM cana-282 122 30 ) ) PUNCT cana-282 122 31 ∫ ∫ PROPN cana-282 122 32 𝑓(π‘₯)𝑑π‘₯ 𝑓(π‘₯)𝑑π‘₯ PROPN cana-282 122 33 𝑏 𝑏 NOUN cana-282 122 34 π‘Ž π‘Ž NOUN cana-282 122 35 = = SYM cana-282 122 36 ∫ ∫ NOUN cana-282 122 37 𝑝1(x)dx 𝑝1(x)dx VERB cana-282 122 38 𝑏 𝑏 PROPN cana-282 122 39 π‘Ž π‘Ž NOUN cana-282 122 40 = = PUNCT cana-282 122 41 β„Ž β„Ž PART cana-282 122 42 2 2 NUM cana-282 122 43 [ [ X cana-282 122 44 𝑦0 𝑦0 NOUN cana-282 122 45 + + CCONJ cana-282 122 46 𝑦1 𝑦1 NOUN cana-282 122 47 ] ] PUNCT cana-282 122 48 repeat repeat VERB cana-282 122 49 the the DET cana-282 122 50 above above ADJ cana-282 122 51 process process NOUN cana-282 122 52 n n PROPN cana-282 122 53 times times PROPN cana-282 122 54 & & CCONJ cana-282 122 55 adding adding PROPN cana-282 122 56 , , PUNCT cana-282 122 57 we we PRON cana-282 122 58 get get VERB cana-282 122 59 this this PRON cana-282 122 60 is be AUX cana-282 122 61 known know VERB cana-282 122 62 as as ADP cana-282 122 63 trapezoidal trapezoidal ADJ cana-282 122 64 rule rule NOUN cana-282 122 65 . . PUNCT cana-282 123 1 modified modify VERB cana-282 123 2 trapezoidal trapezoidal ADJ cana-282 123 3 rule rule NOUN cana-282 123 4 is be AUX cana-282 123 5 based base VERB cana-282 123 6 on on ADP cana-282 123 7 arithmetic arithmetic PROPN cana-282 123 8 & & CCONJ cana-282 123 9 heronian heronian ADJ cana-282 123 10 mean mean PROPN cana-282 124 1 and and CCONJ cana-282 124 2 it it PRON cana-282 124 3 is be AUX cana-282 124 4 the the DET cana-282 124 5 two two NUM cana-282 124 6 arithmetic arithmetic ADJ cana-282 124 7 & & CCONJ cana-282 124 8 heronian heronian ADJ cana-282 124 9 mean mean PROPN cana-282 124 10 respectively respectively ADV cana-282 124 11 . . PUNCT cana-282 125 1 firstly firstly ADV cana-282 125 2 , , PUNCT cana-282 125 3 we we PRON cana-282 125 4 tabulate tabulate VERB cana-282 125 5 the the DET cana-282 125 6 function function NOUN cana-282 125 7 f f PROPN cana-282 125 8 by by ADP cana-282 125 9 taking take VERB cana-282 125 10 h h NOUN cana-282 125 11 = = PUNCT cana-282 126 1 π‘βˆ’π‘Ž π‘βˆ’π‘Ž ADP cana-282 126 2 𝑛 𝑛 ADP cana-282 126 3 x x SYM cana-282 126 4 y y NOUN cana-282 126 5 x0 x0 PROPN cana-282 126 6 y0 y0 PROPN cana-282 126 7 ∫ ∫ NOUN cana-282 126 8 𝑓(π‘₯)𝑑π‘₯ 𝑓(π‘₯)𝑑π‘₯ VERB cana-282 126 9 𝑏 𝑏 NOUN cana-282 126 10 π‘Ž π‘Ž PROPN cana-282 126 11 = = SYM cana-282 126 12 ∫ ∫ NOUN cana-282 126 13 𝑝𝑛(x)dx 𝑝𝑛(x)dx NOUN cana-282 126 14 𝑏 𝑏 NOUN cana-282 126 15 π‘Ž π‘Ž PROPN cana-282 126 16 = = PUNCT cana-282 126 17 βˆ‘ βˆ‘ PROPN cana-282 126 18 π΄π‘˜π‘“(π‘₯π‘˜)𝑛 π΄π‘˜π‘“(π‘₯π‘˜)𝑛 PROPN cana-282 126 19 π‘˜=0 π‘˜=0 ADP cana-282 126 20 … … PUNCT cana-282 126 21 ( ( PUNCT cana-282 126 22 2 2 NUM cana-282 126 23 ) ) PUNCT cana-282 127 1 where where SCONJ cana-282 127 2 π΄π‘˜ π΄π‘˜ PROPN cana-282 127 3 = = PUNCT cana-282 127 4 ∫ ∫ PROPN cana-282 127 5 πΏπ‘˜(π‘₯)𝑑π‘₯ πΏπ‘˜(π‘₯)𝑑π‘₯ X cana-282 128 1 𝑏 𝑏 ADP cana-282 128 2 π‘Ž π‘Ž PROPN cana-282 128 3 ∫ ∫ PROPN cana-282 128 4 𝑓(π‘₯)𝑑π‘₯ 𝑓(π‘₯)𝑑π‘₯ NOUN cana-282 128 5 𝑏 𝑏 NOUN cana-282 128 6 π‘Ž π‘Ž NOUN cana-282 128 7 = = PUNCT cana-282 128 8 β„Ž β„Ž X cana-282 128 9 2 2 NUM cana-282 128 10 [ [ X cana-282 128 11 𝑦0 𝑦0 NOUN cana-282 128 12 + + CCONJ cana-282 128 13 2(𝑦1 2(𝑦1 PROPN cana-282 129 1 + + CCONJ cana-282 129 2 𝑦2 𝑦2 PROPN cana-282 129 3 + + CCONJ cana-282 129 4 … … PUNCT cana-282 129 5 .. .. PUNCT cana-282 129 6 + + SYM cana-282 129 7 π‘¦π‘›βˆ’1 π‘¦π‘›βˆ’1 NOUN cana-282 129 8 ) ) PUNCT cana-282 130 1 + + CCONJ cana-282 130 2 𝑦𝑛 𝑦𝑛 NOUN cana-282 130 3 ] ] X cana-282 130 4 communications communication NOUN cana-282 130 5 on on ADP cana-282 130 6 applied apply VERB cana-282 130 7 nonlinear nonlinear ADJ cana-282 130 8 analysis analysis NOUN cana-282 130 9 issn issn NOUN cana-282 130 10 : : PUNCT cana-282 130 11 1074 1074 NUM cana-282 130 12 - - PUNCT cana-282 130 13 133x 133x NUM cana-282 130 14 vol vol NOUN cana-282 130 15 30 30 NUM cana-282 130 16 no no NOUN cana-282 130 17 . . NOUN cana-282 130 18 4 4 NUM cana-282 130 19 ( ( PUNCT cana-282 130 20 2023 2023 NUM cana-282 130 21 ) ) PUNCT cana-282 131 1 45 45 NUM cana-282 131 2 https://internationalpubls.com https://internationalpubls.com X cana-282 131 3 x1 x1 PUNCT cana-282 131 4 y1 y1 INTJ cana-282 131 5 x2 x2 PROPN cana-282 132 1 y2 y2 PROPN cana-282 132 2 . . PUNCT cana-282 132 3 . . PUNCT cana-282 132 4 . . PUNCT cana-282 132 5 . . PUNCT cana-282 132 6 . . PUNCT cana-282 132 7 . . PUNCT cana-282 132 8 xn-1 xn-1 PUNCT cana-282 133 1 yn-1 yn-1 PUNCT cana-282 134 1 xn xn PROPN cana-282 134 2 yn yn PROPN cana-282 134 3 𝐼1= 𝐼1= ADV cana-282 134 4 ∫ ∫ PROPN cana-282 134 5 𝑓(π‘₯)𝑑π‘₯ 𝑓(π‘₯)𝑑π‘₯ PROPN cana-282 134 6 π‘₯1 π‘₯1 NOUN cana-282 134 7 π‘₯0 π‘₯0 NOUN cana-282 134 8 = = PUNCT cana-282 134 9 β„Ž β„Ž X cana-282 134 10 2 2 NUM cana-282 134 11 ( ( PUNCT cana-282 134 12 𝑦0 𝑦0 NOUN cana-282 134 13 + + CCONJ cana-282 134 14 𝑦1 𝑦1 NOUN cana-282 134 15 ) ) PUNCT cana-282 134 16 𝐼2= 𝐼2= NOUN cana-282 134 17 ∫ ∫ PROPN cana-282 134 18 𝑓(π‘₯)𝑑π‘₯ 𝑓(π‘₯)𝑑π‘₯ PROPN cana-282 134 19 π‘₯2 π‘₯2 PROPN cana-282 134 20 π‘₯1 π‘₯1 NOUN cana-282 134 21 = = SYM cana-282 134 22 β„Ž β„Ž X cana-282 134 23 2 2 NUM cana-282 134 24 ( ( PUNCT cana-282 134 25 𝑦1 𝑦1 PROPN cana-282 134 26 + + CCONJ cana-282 134 27 𝑦2 𝑦2 NOUN cana-282 134 28 ) ) PUNCT cana-282 134 29 𝐼3= 𝐼3= PROPN cana-282 134 30 ∫ ∫ PROPN cana-282 134 31 𝑓(π‘₯)𝑑π‘₯ 𝑓(π‘₯)𝑑π‘₯ PROPN cana-282 134 32 π‘₯3 π‘₯3 NOUN cana-282 134 33 π‘₯2 π‘₯2 NOUN cana-282 134 34 = = PUNCT cana-282 134 35 β„Ž β„Ž X cana-282 134 36 3 3 NUM cana-282 134 37 ( ( PUNCT cana-282 134 38 𝑦2 𝑦2 NOUN cana-282 134 39 + + CCONJ cana-282 134 40 βˆšπ‘¦2𝑦3 βˆšπ‘¦2𝑦3 PROPN cana-282 134 41 + + CCONJ cana-282 134 42 𝑦3 𝑦3 PROPN cana-282 134 43 ) ) PUNCT cana-282 134 44 𝐼4= 𝐼4= PROPN cana-282 134 45 ∫ ∫ PROPN cana-282 134 46 𝑓(π‘₯)𝑑π‘₯ 𝑓(π‘₯)𝑑π‘₯ PROPN cana-282 134 47 π‘₯4 π‘₯4 PROPN cana-282 134 48 π‘₯3 π‘₯3 NOUN cana-282 134 49 = = PUNCT cana-282 134 50 β„Ž β„Ž X cana-282 134 51 3 3 NUM cana-282 134 52 ( ( PUNCT cana-282 134 53 𝑦3 𝑦3 PROPN cana-282 134 54 + + CCONJ cana-282 134 55 βˆšπ‘¦3𝑦4 βˆšπ‘¦3𝑦4 PROPN cana-282 134 56 + + CCONJ cana-282 134 57 𝑦4 𝑦4 NOUN cana-282 134 58 ) ) PUNCT cana-282 134 59 𝐼5= 𝐼5= NOUN cana-282 134 60 ∫ ∫ PROPN cana-282 134 61 𝑓(π‘₯)𝑑π‘₯ 𝑓(π‘₯)𝑑π‘₯ PROPN cana-282 134 62 π‘₯5 π‘₯5 PROPN cana-282 134 63 π‘₯4 π‘₯4 NOUN cana-282 134 64 = = PUNCT cana-282 134 65 β„Ž β„Ž X cana-282 134 66 2 2 NUM cana-282 134 67 ( ( PUNCT cana-282 134 68 𝑦4 𝑦4 NOUN cana-282 134 69 + + CCONJ cana-282 134 70 𝑦5 𝑦5 NOUN cana-282 134 71 ) ) PUNCT cana-282 134 72 continuing continue VERB cana-282 134 73 like like ADP cana-282 134 74 this this PRON cana-282 134 75 and and CCONJ cana-282 134 76 adding add VERB cana-282 134 77 , , PUNCT cana-282 134 78 we we PRON cana-282 134 79 get get VERB cana-282 134 80 this this PRON cana-282 134 81 is be AUX cana-282 134 82 called call VERB cana-282 134 83 modified modified ADJ cana-282 134 84 trapezoidal trapezoidal ADJ cana-282 134 85 rule rule NOUN cana-282 134 86 . . PUNCT cana-282 135 1 3 3 X cana-282 135 2 . . X cana-282 135 3 numerical numerical ADJ cana-282 135 4 examples example NOUN cana-282 135 5 the the DET cana-282 135 6 efficiency efficiency NOUN cana-282 135 7 of of ADP cana-282 135 8 original original ADJ cana-282 135 9 trapezoidal trapezoidal ADJ cana-282 135 10 rule rule NOUN cana-282 135 11 , , PUNCT cana-282 135 12 the the DET cana-282 135 13 modified modify VERB cana-282 135 14 trapezoidal trapezoidal ADJ cana-282 135 15 rule rule NOUN cana-282 135 16 - - PUNCT cana-282 135 17 based base VERB cana-282 135 18 on on ADP cana-282 135 19 arithmetic arithmetic ADJ cana-282 135 20 , , PUNCT cana-282 135 21 & & CCONJ cana-282 135 22 heronian heronian ADJ cana-282 135 23 mean mean PROPN cana-282 135 24 are be AUX cana-282 135 25 contrasted contrast VERB cana-282 135 26 in in ADP cana-282 135 27 this this DET cana-282 135 28 section section NOUN cana-282 135 29 . . PUNCT cana-282 136 1 the the DET cana-282 136 2 integral integral ADJ cana-282 136 3 ∫ ∫ PROPN cana-282 136 4 1 1 NUM cana-282 136 5 1+π‘₯2 1+π‘₯2 PROPN cana-282 136 6 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 136 7 , , PUNCT cana-282 136 8 2 2 NUM cana-282 136 9 1 1 NUM cana-282 136 10 ∫ ∫ NOUN cana-282 136 11 1 1 NUM cana-282 136 12 1+π‘₯ 1+π‘₯ PROPN cana-282 136 13 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 136 14 , , PUNCT cana-282 136 15 4 4 NUM cana-282 136 16 2 2 NUM cana-282 136 17 ∫ ∫ NOUN cana-282 136 18 𝑒π‘₯3 𝑒π‘₯3 PROPN cana-282 136 19 1 1 NUM cana-282 136 20 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 136 21 , , PUNCT cana-282 136 22 ∫ ∫ PROPN cana-282 136 23 log log VERB cana-282 136 24 π‘₯ π‘₯ NOUN cana-282 136 25 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 136 26 , , PUNCT cana-282 136 27 5 5 NUM cana-282 136 28 3 3 NUM cana-282 136 29 ∫ ∫ NOUN cana-282 136 30 2π‘₯2 2π‘₯2 NUM cana-282 136 31 1 1 NUM cana-282 136 32 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 136 33 , , PUNCT cana-282 136 34 ∫ ∫ PROPN cana-282 136 35 sec sec PROPN cana-282 137 1 π‘₯ π‘₯ PROPN cana-282 137 2 tan tan NOUN cana-282 137 3 π‘₯ π‘₯ X cana-282 137 4 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 137 5 , , PUNCT cana-282 137 6 1 1 NUM cana-282 137 7 0 0 NUM cana-282 137 8 and and CCONJ cana-282 137 9 ∫ ∫ PROPN cana-282 137 10 ( ( PUNCT cana-282 137 11 𝑒π‘₯1 𝑒π‘₯1 NOUN cana-282 137 12 0 0 PUNCT cana-282 138 1 + + CCONJ cana-282 138 2 sin sin NOUN cana-282 138 3 π‘₯ π‘₯ NOUN cana-282 138 4 ) ) PUNCT cana-282 138 5 𝑑π‘₯ 𝑑π‘₯ VERB cana-282 138 6 are be AUX cana-282 138 7 evaluated evaluate VERB cana-282 138 8 and and CCONJ cana-282 138 9 results result NOUN cana-282 138 10 are be AUX cana-282 138 11 compared compare VERB cana-282 138 12 . . PUNCT cana-282 139 1 problem problem NOUN cana-282 139 2 1 1 NUM cana-282 139 3 . . PUNCT cana-282 139 4 ∫ ∫ PROPN cana-282 140 1 1 1 NUM cana-282 140 2 1+π‘₯2 1+π‘₯2 PROPN cana-282 140 3 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 140 4 2 2 NUM cana-282 140 5 1 1 NUM cana-282 140 6 actual actual ADJ cana-282 140 7 solution solution NOUN cana-282 140 8 = = NOUN cana-282 140 9 0.3216018366 0.3216018366 VERB cana-282 140 10 by by ADP cana-282 140 11 using use VERB cana-282 140 12 trapezoidal trapezoidal ADJ cana-282 140 13 rule rule NOUN cana-282 140 14 , , PUNCT cana-282 140 15 we we PRON cana-282 140 16 obtained obtain VERB cana-282 140 17 the the DET cana-282 140 18 solution solution NOUN cana-282 140 19 = = NOUN cana-282 140 20 0.34852251407 0.34852251407 NUM cana-282 140 21 by by ADP cana-282 140 22 using use VERB cana-282 140 23 modified modify VERB cana-282 140 24 trapezoidal trapezoidal ADJ cana-282 140 25 rule rule NOUN cana-282 140 26 , , PUNCT cana-282 140 27 we we PRON cana-282 140 28 obtained obtain VERB cana-282 140 29 the the DET cana-282 140 30 solution solution NOUN cana-282 140 31 = = PUNCT cana-282 140 32 0.30315893439 0.30315893439 NUM cana-282 140 33 problem problem NOUN cana-282 140 34 2 2 NUM cana-282 140 35 . . PUNCT cana-282 140 36 ∫ ∫ PROPN cana-282 140 37 1 1 NUM cana-282 141 1 1+π‘₯ 1+π‘₯ NUM cana-282 141 2 𝑑π‘₯ 𝑑π‘₯ ADP cana-282 141 3 4 4 NUM cana-282 141 4 2 2 NUM cana-282 141 5 ∫ ∫ NOUN cana-282 141 6 𝑓(π‘₯)𝑑π‘₯ 𝑓(π‘₯)𝑑π‘₯ VERB cana-282 141 7 𝑏 𝑏 NOUN cana-282 141 8 π‘Ž π‘Ž NOUN cana-282 141 9 = = SYM cana-282 141 10 β„Ž β„Ž NOUN cana-282 141 11 [ [ PUNCT cana-282 141 12 ( ( PUNCT cana-282 141 13 𝑦0 𝑦0 NOUN cana-282 141 14 + + CCONJ cana-282 141 15 𝑦1 𝑦1 NOUN cana-282 141 16 2 2 NUM cana-282 141 17 ) ) PUNCT cana-282 141 18 + + CCONJ cana-282 141 19 ( ( PUNCT cana-282 141 20 𝑦1 𝑦1 NOUN cana-282 141 21 + + CCONJ cana-282 141 22 𝑦2 𝑦2 PROPN cana-282 141 23 2 2 NUM cana-282 141 24 ) ) PUNCT cana-282 141 25 + + CCONJ cana-282 141 26 ( ( PUNCT cana-282 141 27 𝑦2 𝑦2 NOUN cana-282 141 28 + + CCONJ cana-282 141 29 βˆšπ‘¦2𝑦3 βˆšπ‘¦2𝑦3 PROPN cana-282 141 30 + + CCONJ cana-282 141 31 𝑦3 𝑦3 PROPN cana-282 141 32 3 3 NUM cana-282 141 33 ) ) PUNCT cana-282 142 1 + + CCONJ cana-282 142 2 ( ( PUNCT cana-282 142 3 𝑦3 𝑦3 PROPN cana-282 142 4 + + CCONJ cana-282 142 5 βˆšπ‘¦3𝑦4 βˆšπ‘¦3𝑦4 PROPN cana-282 142 6 + + CCONJ cana-282 142 7 𝑦4 𝑦4 NOUN cana-282 142 8 3 3 NUM cana-282 142 9 ) ) PUNCT cana-282 142 10 + + CCONJ cana-282 142 11 β‹― β‹― VERB cana-282 142 12 communications communication NOUN cana-282 142 13 on on ADP cana-282 142 14 applied apply VERB cana-282 142 15 nonlinear nonlinear ADJ cana-282 142 16 analysis analysis NOUN cana-282 142 17 issn issn NOUN cana-282 142 18 : : PUNCT cana-282 142 19 1074 1074 NUM cana-282 142 20 - - PUNCT cana-282 142 21 133x 133x NUM cana-282 142 22 vol vol NOUN cana-282 142 23 30 30 NUM cana-282 142 24 no no NOUN cana-282 142 25 . . NOUN cana-282 142 26 4 4 NUM cana-282 142 27 ( ( PUNCT cana-282 142 28 2023 2023 NUM cana-282 142 29 ) ) PUNCT cana-282 142 30 46 46 NUM cana-282 142 31 https://internationalpubls.com https://internationalpubls.com NUM cana-282 142 32 actual actual ADJ cana-282 142 33 solution solution NOUN cana-282 142 34 = = NOUN cana-282 142 35 0.91629073187 0.91629073187 NUM cana-282 142 36 by by ADP cana-282 142 37 using use VERB cana-282 142 38 trapezoidal trapezoidal ADJ cana-282 142 39 rule rule NOUN cana-282 142 40 , , PUNCT cana-282 142 41 we we PRON cana-282 142 42 obtained obtain VERB cana-282 142 43 the the DET cana-282 142 44 solution solution NOUN cana-282 142 45 = = SYM cana-282 142 46 0.5123015873 0.5123015873 NUM cana-282 142 47 by by ADP cana-282 142 48 using use VERB cana-282 142 49 modified modify VERB cana-282 142 50 trapezoidal trapezoidal ADJ cana-282 142 51 rule rule NOUN cana-282 142 52 , , PUNCT cana-282 142 53 we we PRON cana-282 142 54 obtained obtain VERB cana-282 142 55 the the DET cana-282 142 56 solution solution NOUN cana-282 142 57 = = NOUN cana-282 142 58 0.88718467879 0.88718467879 NUM cana-282 142 59 problem problem NOUN cana-282 142 60 3 3 NUM cana-282 142 61 . . PUNCT cana-282 142 62 ∫ ∫ PROPN cana-282 143 1 𝑒π‘₯3 𝑒π‘₯3 PROPN cana-282 143 2 1 1 NUM cana-282 143 3 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 143 4 actual actual ADJ cana-282 143 5 solution solution NOUN cana-282 143 6 = = NOUN cana-282 143 7 17.3672550947 17.3672550947 NUM cana-282 143 8 by by ADP cana-282 143 9 using use VERB cana-282 143 10 trapezoidal trapezoidal ADJ cana-282 143 11 rule rule NOUN cana-282 143 12 , , PUNCT cana-282 143 13 we we PRON cana-282 143 14 obtained obtain VERB cana-282 143 15 the the DET cana-282 143 16 solution solution NOUN cana-282 143 17 = = NOUN cana-282 143 18 17.7275742529 17.7275742529 NUM cana-282 143 19 by by ADP cana-282 143 20 using use VERB cana-282 143 21 modified modify VERB cana-282 143 22 trapezoidal trapezoidal ADJ cana-282 143 23 rule rule NOUN cana-282 143 24 , , PUNCT cana-282 143 25 we we PRON cana-282 143 26 obtained obtain VERB cana-282 143 27 the the DET cana-282 143 28 solution solution NOUN cana-282 143 29 = = SYM cana-282 143 30 17.5960037893 17.5960037893 NUM cana-282 143 31 problem problem NOUN cana-282 143 32 4 4 NUM cana-282 143 33 . . PUNCT cana-282 144 1 ∫ ∫ PROPN cana-282 144 2 log log VERB cana-282 144 3 π‘₯ π‘₯ NOUN cana-282 144 4 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 144 5 5 5 NUM cana-282 144 6 3 3 NUM cana-282 144 7 actual actual ADJ cana-282 144 8 solution solution NOUN cana-282 144 9 = = NOUN cana-282 144 10 2.75135269617 2.75135269617 NUM cana-282 144 11 by by ADP cana-282 144 12 using use VERB cana-282 144 13 trapezoidal trapezoidal ADJ cana-282 144 14 rule rule NOUN cana-282 144 15 , , PUNCT cana-282 144 16 we we PRON cana-282 144 17 obtained obtain VERB cana-282 144 18 the the DET cana-282 144 19 solution solution NOUN cana-282 144 20 = = NOUN cana-282 144 21 4.12286987021 4.12286987021 NUM cana-282 144 22 by by ADP cana-282 144 23 using use VERB cana-282 144 24 modified modify VERB cana-282 144 25 trapezoidal trapezoidal ADJ cana-282 144 26 rule rule NOUN cana-282 144 27 , , PUNCT cana-282 144 28 we we PRON cana-282 144 29 obtained obtain VERB cana-282 144 30 the the DET cana-282 144 31 solution solution NOUN cana-282 144 32 = = SYM cana-282 144 33 4.12234686636 4.12234686636 NUM cana-282 144 34 problem problem NOUN cana-282 144 35 5 5 NUM cana-282 144 36 . . PUNCT cana-282 144 37 ∫ ∫ PROPN cana-282 145 1 2π‘₯2 2π‘₯2 NUM cana-282 145 2 1 1 NUM cana-282 145 3 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 145 4 actual actual ADJ cana-282 145 5 solution solution NOUN cana-282 145 6 = = NOUN cana-282 145 7 2.88539008178 2.88539008178 NUM cana-282 145 8 by by ADP cana-282 145 9 using use VERB cana-282 145 10 trapezoidal trapezoidal ADJ cana-282 145 11 rule rule NOUN cana-282 145 12 , , PUNCT cana-282 145 13 we we PRON cana-282 145 14 obtained obtain VERB cana-282 145 15 the the DET cana-282 145 16 solution solution NOUN cana-282 145 17 = = SYM cana-282 145 18 2.89260675394 2.89260675394 NUM cana-282 145 19 by by ADP cana-282 145 20 using use VERB cana-282 145 21 modified modify VERB cana-282 145 22 trapezoidal trapezoidal ADJ cana-282 145 23 rule rule NOUN cana-282 145 24 , , PUNCT cana-282 145 25 we we PRON cana-282 145 26 obtained obtain VERB cana-282 145 27 the the DET cana-282 145 28 solution solution NOUN cana-282 145 29 = = NOUN cana-282 145 30 2.8904933039 2.8904933039 NUM cana-282 145 31 problem problem NOUN cana-282 145 32 6 6 NUM cana-282 145 33 . . PUNCT cana-282 146 1 ∫ ∫ PROPN cana-282 147 1 sec sec PROPN cana-282 148 1 π‘₯ π‘₯ PROPN cana-282 148 2 tan tan NOUN cana-282 148 3 π‘₯ π‘₯ NOUN cana-282 148 4 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 148 5 1 1 NUM cana-282 148 6 0 0 NUM cana-282 148 7 actual actual ADJ cana-282 148 8 solution solution NOUN cana-282 148 9 = = NOUN cana-282 148 10 0.85081571768 0.85081571768 NUM cana-282 148 11 by by ADP cana-282 148 12 using use VERB cana-282 148 13 trapezoidal trapezoidal ADJ cana-282 148 14 rule rule NOUN cana-282 148 15 , , PUNCT cana-282 148 16 we we PRON cana-282 148 17 obtained obtain VERB cana-282 148 18 the the DET cana-282 148 19 solution solution NOUN cana-282 148 20 = = NOUN cana-282 148 21 0.90012355497 0.90012355497 NUM cana-282 148 22 by by ADP cana-282 148 23 using use VERB cana-282 148 24 modified modify VERB cana-282 148 25 trapezoidal trapezoidal ADJ cana-282 148 26 rule rule NOUN cana-282 148 27 , , PUNCT cana-282 148 28 we we PRON cana-282 148 29 obtained obtain VERB cana-282 148 30 the the DET cana-282 148 31 solution solution NOUN cana-282 148 32 = = NOUN cana-282 148 33 0.88181471573 0.88181471573 NUM cana-282 148 34 problem7 problem7 NOUN cana-282 148 35 . . PUNCT cana-282 149 1 ∫ ∫ PROPN cana-282 149 2 ( ( PUNCT cana-282 149 3 𝑒π‘₯1 𝑒π‘₯1 NOUN cana-282 149 4 0 0 PUNCT cana-282 150 1 + + CCONJ cana-282 150 2 sin sin NOUN cana-282 150 3 π‘₯ π‘₯ NOUN cana-282 150 4 ) ) PUNCT cana-282 150 5 𝑑π‘₯ 𝑑π‘₯ VERB cana-282 150 6 actual actual ADJ cana-282 150 7 solution solution NOUN cana-282 150 8 = = NOUN cana-282 150 9 2.17797952259 2.17797952259 NUM cana-282 150 10 by by ADP cana-282 150 11 using use VERB cana-282 150 12 trapezoidal trapezoidal ADJ cana-282 150 13 rule rule NOUN cana-282 150 14 , , PUNCT cana-282 150 15 we we PRON cana-282 150 16 obtained obtain VERB cana-282 150 17 the the DET cana-282 150 18 solution solution NOUN cana-282 150 19 = = NOUN cana-282 150 20 2.18452284213 2.18452284213 NUM cana-282 150 21 by by ADP cana-282 150 22 using use VERB cana-282 150 23 modified modify VERB cana-282 150 24 trapezoidal trapezoidal ADJ cana-282 150 25 rule rule NOUN cana-282 150 26 , , PUNCT cana-282 150 27 we we PRON cana-282 150 28 obtained obtain VERB cana-282 150 29 the the DET cana-282 150 30 solution solution NOUN cana-282 150 31 = = NOUN cana-282 150 32 2.1807080442 2.1807080442 NUM cana-282 150 33 table table NOUN cana-282 150 34 1 1 NUM cana-282 150 35 comparison comparison NOUN cana-282 150 36 results result NOUN cana-282 150 37 of of ADP cana-282 150 38 trapezoidal trapezoidal ADJ cana-282 150 39 , , PUNCT cana-282 150 40 modified modify VERB cana-282 150 41 trapezoidal trapezoidal ADJ cana-282 150 42 methods method NOUN cana-282 150 43 are be AUX cana-282 150 44 demonstrated demonstrate VERB cana-282 150 45 in in ADP cana-282 150 46 the the DET cana-282 150 47 table table NOUN cana-282 150 48 given give VERB cana-282 150 49 below below ADP cana-282 150 50 function function NOUN cana-282 150 51 exact exact ADJ cana-282 150 52 value value NOUN cana-282 150 53 trapezoidal trapezoidal ADJ cana-282 150 54 rule rule NOUN cana-282 150 55 error error NOUN cana-282 150 56 modified modify VERB cana-282 150 57 trapezoidal trapezoidal ADJ cana-282 150 58 rule rule NOUN cana-282 150 59 error error NOUN cana-282 150 60 ∫ ∫ NOUN cana-282 150 61 1 1 NUM cana-282 150 62 1+π‘₯2 1+π‘₯2 PROPN cana-282 150 63 𝑑π‘₯ 𝑑π‘₯ NOUN cana-282 150 64 2 2 NUM cana-282 150 65 1 1 NUM cana-282 150 66 0.3216018366 0.3216018366 NUM cana-282 150 67 0.34852251407 0.34852251407 NUM cana-282 150 68 0.0269206775 0.0269206775 NUM cana-282 150 69 0.30315893439 0.30315893439 NUM cana-282 150 70 0.01844290221 0.01844290221 NUM cana-282 150 71 communications communication NOUN cana-282 150 72 on on ADP cana-282 150 73 applied apply VERB cana-282 150 74 nonlinear nonlinear ADJ cana-282 150 75 analysis analysis NOUN cana-282 150 76 issn issn NOUN cana-282 150 77 : : PUNCT cana-282 150 78 1074 1074 NUM cana-282 150 79 - - PUNCT cana-282 150 80 133x 133x NUM cana-282 150 81 vol vol NOUN cana-282 150 82 30 30 NUM cana-282 150 83 no no NOUN cana-282 150 84 . . NOUN cana-282 150 85 4 4 NUM cana-282 150 86 ( ( PUNCT cana-282 150 87 2023 2023 NUM cana-282 150 88 ) ) PUNCT cana-282 151 1 47 47 NUM cana-282 151 2 https://internationalpubls.com https://internationalpubls.com X cana-282 151 3 ∫ ∫ PROPN cana-282 151 4 1 1 NUM cana-282 152 1 1+π‘₯ 1+π‘₯ NUM cana-282 152 2 𝑑π‘₯ 𝑑π‘₯ ADP cana-282 152 3 4 4 NUM cana-282 152 4 2 2 NUM cana-282 152 5 0.91629073187 0.91629073187 NUM cana-282 152 6 0.5123015873 0.5123015873 NUM cana-282 152 7 0.40398914457 0.40398914457 NUM cana-282 152 8 0.88718467879 0.88718467879 NUM cana-282 152 9 0.02910605308 0.02910605308 NUM cana-282 152 10 ∫ ∫ NOUN cana-282 152 11 𝑒π‘₯3 𝑒π‘₯3 ADP cana-282 152 12 1 1 NUM cana-282 152 13 𝑑π‘₯ 𝑑π‘₯ ADP cana-282 152 14 17.3672550947 17.3672550947 NUM cana-282 152 15 17.7275742529 17.7275742529 NUM cana-282 152 16 0.3603191582 0.3603191582 NUM cana-282 152 17 17.5960037893 17.5960037893 NUM cana-282 152 18 0.2287486943 0.2287486943 NUM cana-282 152 19 ∫ ∫ NOUN cana-282 152 20 log log NOUN cana-282 152 21 π‘₯ π‘₯ NOUN cana-282 152 22 𝑑π‘₯ 𝑑π‘₯ VERB cana-282 152 23 5 5 NUM cana-282 152 24 3 3 NUM cana-282 152 25 2.75135269617 2.75135269617 NUM cana-282 152 26 4.12286987021 4.12286987021 NUM cana-282 152 27 1.371517174 1.371517174 NUM cana-282 152 28 4.12234686636 4.12234686636 NUM cana-282 152 29 1.3709941702 1.3709941702 NUM cana-282 152 30 ∫ ∫ NOUN cana-282 152 31 2π‘₯2 2π‘₯2 NUM cana-282 152 32 1 1 NUM cana-282 152 33 𝑑π‘₯ 𝑑π‘₯ ADP cana-282 152 34 2.88539008178 2.88539008178 NUM cana-282 152 35 2.89260675394 2.89260675394 NUM cana-282 152 36 0.0072166722 0.0072166722 NUM cana-282 152 37 2.8904933039 2.8904933039 NUM cana-282 152 38 0.0051032221 0.0051032221 NUM cana-282 152 39 ∫ ∫ NOUN cana-282 152 40 sec sec PROPN cana-282 152 41 π‘₯ π‘₯ PROPN cana-282 152 42 tan tan NOUN cana-282 152 43 π‘₯ π‘₯ INTJ cana-282 152 44 𝑑π‘₯ 𝑑π‘₯ VERB cana-282 152 45 1 1 NUM cana-282 152 46 0 0 NUM cana-282 152 47 0.85081571768 0.85081571768 NUM cana-282 152 48 0.90012355497 0.90012355497 NUM cana-282 152 49 0.0493078373 0.0493078373 NUM cana-282 152 50 0.88181471573 0.88181471573 NUM cana-282 152 51 0.0309989981 0.0309989981 NUM cana-282 152 52 ∫ ∫ NOUN cana-282 152 53 ( ( PUNCT cana-282 152 54 𝑒π‘₯1 𝑒π‘₯1 NOUN cana-282 152 55 0 0 PUNCT cana-282 153 1 + + CCONJ cana-282 153 2 sin sin NOUN cana-282 153 3 π‘₯ π‘₯ NOUN cana-282 153 4 ) ) PUNCT cana-282 153 5 𝑑π‘₯ 𝑑π‘₯ VERB cana-282 153 6 2.17797952259 2.17797952259 NUM cana-282 153 7 2.18452284213 2.18452284213 NUM cana-282 153 8 0.0065433195 0.0065433195 NUM cana-282 153 9 2.1807080442 2.1807080442 NUM cana-282 153 10 0.0027285216 0.0027285216 NUM cana-282 153 11 4 4 NUM cana-282 153 12 . . PUNCT cana-282 153 13 result result NOUN cana-282 153 14 discussion discussion NOUN cana-282 153 15 by by ADP cana-282 153 16 concentrating concentrate VERB cana-282 153 17 on on ADP cana-282 153 18 the the DET cana-282 153 19 preceding precede VERB cana-282 153 20 results result NOUN cana-282 153 21 , , PUNCT cana-282 153 22 we we PRON cana-282 153 23 moved move VERB cana-282 153 24 forward forward ADV cana-282 153 25 with with ADP cana-282 153 26 the the DET cana-282 153 27 interpretation interpretation NOUN cana-282 153 28 of of ADP cana-282 153 29 our our PRON cana-282 153 30 numerical numerical ADJ cana-282 153 31 approaches approach NOUN cana-282 153 32 for for ADP cana-282 153 33 the the DET cana-282 153 34 approximation approximation NOUN cana-282 153 35 solutions solution NOUN cana-282 153 36 of of ADP cana-282 153 37 non non ADJ cana-282 153 38 - - ADJ cana-282 153 39 linear linear ADJ cana-282 153 40 equations equation NOUN cana-282 153 41 . . PUNCT cana-282 154 1 problems problem NOUN cana-282 154 2 are be AUX cana-282 154 3 solved solve VERB cana-282 154 4 by by ADP cana-282 154 5 using use VERB cana-282 154 6 trapezoidal trapezoidal ADJ cana-282 154 7 and and CCONJ cana-282 154 8 modified modify VERB cana-282 154 9 trapezoidal trapezoidal ADJ cana-282 154 10 methods method NOUN cana-282 154 11 . . PUNCT cana-282 155 1 the the DET cana-282 155 2 results result NOUN cana-282 155 3 table table NOUN cana-282 155 4 indicates indicate VERB cana-282 155 5 that that SCONJ cana-282 155 6 compared compare VERB cana-282 155 7 to to ADP cana-282 155 8 the the DET cana-282 155 9 trapezoidal trapezoidal ADJ cana-282 155 10 approaches approach NOUN cana-282 155 11 , , PUNCT cana-282 155 12 the the DET cana-282 155 13 modified modify VERB cana-282 155 14 trapezoidal trapezoidal ADJ cana-282 155 15 methodology methodology NOUN cana-282 155 16 converges converge VERB cana-282 155 17 to to ADP cana-282 155 18 the the DET cana-282 155 19 exact exact ADJ cana-282 155 20 answer answer NOUN cana-282 155 21 more more ADV cana-282 155 22 frequently frequently ADV cana-282 155 23 . . PUNCT cana-282 156 1 the the DET cana-282 156 2 modified modify VERB cana-282 156 3 trapezoidal trapezoidal ADJ cana-282 156 4 technique technique NOUN cana-282 156 5 has have VERB cana-282 156 6 lowest low ADJ cana-282 156 7 error error NOUN cana-282 156 8 & & CCONJ cana-282 156 9 highest high ADJ cana-282 156 10 error error NOUN cana-282 156 11 for for ADP cana-282 156 12 the the DET cana-282 156 13 trapezoidal trapezoidal ADJ cana-282 156 14 technique technique NOUN cana-282 156 15 . . PUNCT cana-282 157 1 in in ADP cana-282 157 2 conclusion conclusion NOUN cana-282 157 3 , , PUNCT cana-282 157 4 it it PRON cana-282 157 5 has have AUX cana-282 157 6 been be AUX cana-282 157 7 observed observe VERB cana-282 157 8 that that SCONJ cana-282 157 9 the the DET cana-282 157 10 modified modify VERB cana-282 157 11 trapezoidal trapezoidal ADJ cana-282 157 12 approach approach NOUN cana-282 157 13 is be AUX cana-282 157 14 the the DET cana-282 157 15 most most ADV cana-282 157 16 effective effective ADJ cana-282 157 17 way way NOUN cana-282 157 18 for for ADP cana-282 157 19 solving solve VERB cana-282 157 20 the the DET cana-282 157 21 nonlinear nonlinear ADJ cana-282 157 22 equations equation NOUN cana-282 157 23 since since SCONJ cana-282 157 24 it it PRON cana-282 157 25 converges converge VERB cana-282 157 26 more more ADV cana-282 157 27 precisely precisely ADV cana-282 157 28 and and CCONJ cana-282 157 29 fast fast ADJ cana-282 157 30 than than ADP cana-282 157 31 the the DET cana-282 157 32 trapezoidal trapezoidal ADJ cana-282 157 33 methods method NOUN cana-282 157 34 . . PUNCT cana-282 158 1 5 5 X cana-282 158 2 . . X cana-282 158 3 conclusion conclusion NOUN cana-282 158 4 we we PRON cana-282 158 5 are be AUX cana-282 158 6 capable capable ADJ cana-282 158 7 of of ADP cana-282 158 8 demonstrating demonstrate VERB cana-282 158 9 numerical numerical ADJ cana-282 158 10 integration integration NOUN cana-282 158 11 with with ADP cana-282 158 12 the the DET cana-282 158 13 methods method NOUN cana-282 158 14 of of ADP cana-282 158 15 trapezoidal trapezoidal ADJ cana-282 158 16 and and CCONJ cana-282 158 17 modified modify VERB cana-282 158 18 trapezoidal trapezoidal NOUN cana-282 158 19 that that SCONJ cana-282 158 20 we we PRON cana-282 158 21 described describe VERB cana-282 158 22 from from ADP cana-282 158 23 the the DET cana-282 158 24 approaches approach NOUN cana-282 158 25 covered cover VERB cana-282 158 26 in in ADP cana-282 158 27 our our PRON cana-282 158 28 research research NOUN cana-282 158 29 in in ADP cana-282 158 30 order order NOUN cana-282 158 31 to to PART cana-282 158 32 get get VERB cana-282 158 33 the the DET cana-282 158 34 minimum minimum ADJ cana-282 158 35 error error NOUN cana-282 158 36 value value NOUN cana-282 158 37 . . PUNCT cana-282 159 1 in in ADP cana-282 159 2 our our PRON cana-282 159 3 written write VERB cana-282 159 4 assignments assignment NOUN cana-282 159 5 , , PUNCT cana-282 159 6 we we PRON cana-282 159 7 have have AUX cana-282 159 8 made make VERB cana-282 159 9 an an DET cana-282 159 10 effort effort NOUN cana-282 159 11 to to PART cana-282 159 12 illustrate illustrate VERB cana-282 159 13 certain certain ADJ cana-282 159 14 examples example NOUN cana-282 159 15 and and CCONJ cana-282 159 16 highlight highlight VERB cana-282 159 17 the the DET cana-282 159 18 circumstances circumstance NOUN cana-282 159 19 in in ADP cana-282 159 20 which which PRON cana-282 159 21 the the DET cana-282 159 22 modified modify VERB cana-282 159 23 trapezoidal trapezoidal ADJ cana-282 159 24 rule rule NOUN cana-282 159 25 works work VERB cana-282 159 26 well well ADV cana-282 159 27 . . PUNCT cana-282 160 1 as as ADP cana-282 160 2 a a DET cana-282 160 3 result result NOUN cana-282 160 4 , , PUNCT cana-282 160 5 we we PRON cana-282 160 6 can can AUX cana-282 160 7 see see VERB cana-282 160 8 that that SCONJ cana-282 160 9 the the DET cana-282 160 10 modified modify VERB cana-282 160 11 trapezoidal trapezoidal ADJ cana-282 160 12 rule rule NOUN cana-282 160 13 provides provide VERB cana-282 160 14 the the DET cana-282 160 15 lowest low ADJ cana-282 160 16 error error NOUN cana-282 160 17 value value NOUN cana-282 160 18 and and CCONJ cana-282 160 19 is be AUX cana-282 160 20 , , PUNCT cana-282 160 21 formally formally ADV cana-282 160 22 , , PUNCT cana-282 160 23 the the DET cana-282 160 24 most most ADV cana-282 160 25 practical practical ADJ cana-282 160 26 and and CCONJ cana-282 160 27 appropriate appropriate ADJ cana-282 160 28 method method NOUN cana-282 160 29 . . PUNCT cana-282 161 1 references reference NOUN cana-282 161 2 [ [ X cana-282 161 3 1 1 NUM cana-282 161 4 ] ] X cana-282 161 5 s.s s.s PROPN cana-282 161 6 sastry sastry PROPN cana-282 161 7 , , PUNCT cana-282 161 8 2007 2007 NUM cana-282 161 9 , , PUNCT cana-282 161 10 introductory introductory ADJ cana-282 161 11 method method NOUN cana-282 161 12 of of ADP cana-282 161 13 numerical numerical ADJ cana-282 161 14 analysis analysis NOUN cana-282 161 15 , , PUNCT cana-282 161 16 fourth fourth ADJ cana-282 161 17 edition edition NOUN cana-282 161 18 , , PUNCT cana-282 161 19 prentice prentice NOUN cana-282 161 20 - - PUNCT cana-282 161 21 hall hall NOUN cana-282 161 22 of of ADP cana-282 161 23 india india PROPN cana-282 161 24 private private PROPN cana-282 161 25 limited limited ADJ cana-282 161 26 . . PUNCT cana-282 162 1 [ [ X cana-282 162 2 2 2 NUM cana-282 162 3 ] ] PUNCT cana-282 162 4 richard richard PROPN cana-282 162 5 l. l. PROPN cana-282 162 6 burden burden PROPN cana-282 162 7 , , PUNCT cana-282 162 8 2007 2007 NUM cana-282 162 9 , , PUNCT cana-282 162 10 numerical numerical ADJ cana-282 162 11 analysis analysis NOUN cana-282 162 12 , , PUNCT cana-282 162 13 seven seven NUM cana-282 162 14 edition edition NOUN cana-282 162 15 , , PUNCT cana-282 162 16 international international PROPN cana-282 162 17 thomson thomson PROPN cana-282 162 18 publishing publishing PROPN cana-282 162 19 company company NOUN cana-282 162 20 . . PUNCT cana-282 163 1 [ [ X cana-282 163 2 3 3 X cana-282 163 3 ] ] PUNCT cana-282 163 4 jonh jonh PROPN cana-282 163 5 h. h. PROPN cana-282 163 6 mathew mathew PROPN cana-282 163 7 , , PUNCT cana-282 163 8 2000 2000 NUM cana-282 163 9 , , PUNCT cana-282 163 10 numerical numerical ADJ cana-282 163 11 method method NOUN cana-282 163 12 for for ADP cana-282 163 13 mathematics mathematic NOUN cana-282 163 14 , , PUNCT cana-282 163 15 science science NOUN cana-282 163 16 and and CCONJ cana-282 163 17 engineering engineering NOUN cana-282 163 18 , , PUNCT cana-282 163 19 second second ADJ cana-282 163 20 edition edition NOUN cana-282 163 21 , , PUNCT cana-282 163 22 prentice prentice NOUN cana-282 163 23 hall hall NOUN cana-282 163 24 of of ADP cana-282 163 25 india india PROPN cana-282 163 26 private private PROPN cana-282 163 27 limited limited ADJ cana-282 163 28 . . PUNCT cana-282 164 1 [ [ X cana-282 164 2 4 4 X cana-282 164 3 ] ] PUNCT cana-282 164 4 j. j. PROPN cana-282 164 5 oliver oliver PROPN cana-282 164 6 , , PUNCT cana-282 164 7 1971 1971 NUM cana-282 164 8 , , PUNCT cana-282 164 9 the the DET cana-282 164 10 evaluation evaluation NOUN cana-282 164 11 of of ADP cana-282 164 12 definite definite ADJ cana-282 164 13 integrals integral NOUN cana-282 164 14 using use VERB cana-282 164 15 high high ADJ cana-282 164 16 - - PUNCT cana-282 164 17 order order NOUN cana-282 164 18 formulae formulae NOUN cana-282 164 19 , , PUNCT cana-282 164 20 the the DET cana-282 164 21 computer computer NOUN cana-282 164 22 journal journal NOUN cana-282 164 23 , , PUNCT cana-282 164 24 vol-14(3 vol-14(3 PROPN cana-282 164 25 ) ) PUNCT cana-282 164 26 . . PUNCT cana-282 165 1 [ [ X cana-282 165 2 5 5 NUM cana-282 165 3 ] ] PUNCT cana-282 165 4 gerry gerry PROPN cana-282 165 5 sozio sozio PROPN cana-282 165 6 , , PUNCT cana-282 165 7 2009 2009 NUM cana-282 165 8 , , PUNCT cana-282 165 9 numerical numerical ADJ cana-282 165 10 integration integration NOUN cana-282 165 11 , , PUNCT cana-282 165 12 australian australian ADJ cana-282 165 13 senior senior ADJ cana-282 165 14 mathematics mathematic NOUN cana-282 165 15 journal journal NOUN cana-282 165 16 , , PUNCT cana-282 165 17 vol-23(1 vol-23(1 PROPN cana-282 165 18 ) ) PUNCT cana-282 165 19 . . PUNCT cana-282 166 1 [ [ X cana-282 166 2 6 6 NUM cana-282 166 3 ] ] X cana-282 166 4 evans evans PROPN cana-282 166 5 m m PROPN cana-282 166 6 , , PUNCT cana-282 166 7 swartz swartz PROPN cana-282 166 8 t. t. PROPN cana-282 166 9 methods method NOUN cana-282 166 10 for for ADP cana-282 166 11 approximating approximate VERB cana-282 166 12 integrals integral NOUN cana-282 166 13 in in ADP cana-282 166 14 statistics statistic NOUN cana-282 166 15 with with ADP cana-282 166 16 special special ADJ cana-282 166 17 emphasis emphasis NOUN cana-282 166 18 on on ADP cana-282 166 19 bayesian bayesian ADJ cana-282 166 20 integration integration NOUN cana-282 166 21 problems problem NOUN cana-282 166 22 . . PUNCT cana-282 167 1 statistical statistical ADJ cana-282 167 2 science science NOUN cana-282 167 3 . . PUNCT cana-282 168 1 19995;10 19995;10 NUM cana-282 168 2 : : PUNCT cana-282 168 3 254 254 NUM cana-282 168 4 - - SYM cana-282 168 5 272 272 NUM cana-282 168 6 . . PUNCT cana-282 169 1 communications communication NOUN cana-282 169 2 on on ADP cana-282 169 3 applied apply VERB cana-282 169 4 nonlinear nonlinear ADJ cana-282 169 5 analysis analysis NOUN cana-282 169 6 issn issn NOUN cana-282 169 7 : : PUNCT cana-282 169 8 1074 1074 NUM cana-282 169 9 - - PUNCT cana-282 169 10 133x 133x NUM cana-282 169 11 vol vol NOUN cana-282 169 12 30 30 NUM cana-282 169 13 no no NOUN cana-282 169 14 . . NOUN cana-282 169 15 4 4 NUM cana-282 169 16 ( ( PUNCT cana-282 169 17 2023 2023 NUM cana-282 169 18 ) ) PUNCT cana-282 169 19 48 48 NUM cana-282 169 20 https://internationalpubls.com https://internationalpubls.com X cana-282 170 1 [ [ X cana-282 170 2 7 7 NUM cana-282 170 3 ] ] PUNCT cana-282 170 4 ohta ohta NOUN cana-282 170 5 , , PUNCT cana-282 170 6 koji koji PROPN cana-282 170 7 , , PUNCT cana-282 170 8 and and CCONJ cana-282 170 9 hatsuo hatsuo PROPN cana-282 170 10 ishida ishida PROPN cana-282 170 11 . . PUNCT cana-282 171 1 " " PUNCT cana-282 171 2 comparison comparison NOUN cana-282 171 3 among among ADP cana-282 171 4 several several ADJ cana-282 171 5 numerical numerical ADJ cana-282 171 6 integration integration NOUN cana-282 171 7 methods method NOUN cana-282 171 8 for for ADP cana-282 171 9 kramers kramer NOUN cana-282 171 10 - - PUNCT cana-282 171 11 kronig kronig NOUN cana-282 171 12 transformation transformation NOUN cana-282 171 13 . . PUNCT cana-282 171 14 " " PUNCT cana-282 171 15 applied apply VERB cana-282 171 16 spectroscopy spectroscopy NOUN cana-282 171 17 42.6 42.6 NUM cana-282 171 18 ( ( PUNCT cana-282 171 19 1988 1988 NUM cana-282 171 20 ): ): PUNCT cana-282 171 21 952 952 NUM cana-282 171 22 - - SYM cana-282 171 23 957 957 NUM cana-282 171 24 . . PUNCT cana-282 172 1 [ [ X cana-282 172 2 8 8 NUM cana-282 172 3 ] ] SYM cana-282 172 4 pennestrΓ¬ pennestrΓ¬ ADJ cana-282 172 5 , , PUNCT cana-282 172 6 ettore ettore NOUN cana-282 172 7 , , PUNCT cana-282 172 8 valerio valerio PROPN cana-282 172 9 rossi rossi PROPN cana-282 172 10 , , PUNCT cana-282 172 11 pietro pietro PROPN cana-282 172 12 salvini salvini PROPN cana-282 172 13 , , PUNCT cana-282 172 14 and and CCONJ cana-282 172 15 pier pier PROPN cana-282 172 16 paolo paolo PROPN cana-282 172 17 valentini valentini PROPN cana-282 172 18 . . PUNCT cana-282 173 1 " " PUNCT cana-282 173 2 review review NOUN cana-282 173 3 and and CCONJ cana-282 173 4 comparison comparison NOUN cana-282 173 5 of of ADP cana-282 173 6 dry dry ADJ cana-282 173 7 friction friction NOUN cana-282 173 8 force force NOUN cana-282 173 9 models model NOUN cana-282 173 10 . . PUNCT cana-282 173 11 " " PUNCT cana-282 174 1 nonlinear nonlinear ADJ cana-282 174 2 dynamics dynamic NOUN cana-282 174 3 83 83 NUM cana-282 174 4 , , PUNCT cana-282 174 5 no no INTJ cana-282 174 6 . . NOUN cana-282 174 7 4 4 NUM cana-282 174 8 ( ( PUNCT cana-282 174 9 2016 2016 NUM cana-282 174 10 ): ): PUNCT cana-282 174 11 1785 1785 NUM cana-282 174 12 - - SYM cana-282 174 13 1801 1801 NUM cana-282 174 14 . . PUNCT cana-282 175 1 [ [ X cana-282 175 2 9 9 NUM cana-282 175 3 ] ] PUNCT cana-282 175 4 uilhoorn uilhoorn VERB cana-282 175 5 , , PUNCT cana-282 175 6 ferdinand ferdinand PROPN cana-282 175 7 evert evert PROPN cana-282 175 8 . . PUNCT cana-282 176 1 " " PUNCT cana-282 176 2 a a DET cana-282 176 3 comparison comparison NOUN cana-282 176 4 of of ADP cana-282 176 5 numerical numerical ADJ cana-282 176 6 integration integration NOUN cana-282 176 7 schemes scheme NOUN cana-282 176 8 for for ADP cana-282 176 9 particle particle NOUN cana-282 176 10 filterbased filterbase VERB cana-282 176 11 estimation estimation NOUN cana-282 176 12 of of ADP cana-282 176 13 gas gas NOUN cana-282 176 14 flow flow NOUN cana-282 176 15 dynamics dynamic NOUN cana-282 176 16 . . PUNCT cana-282 176 17 " " PUNCT cana-282 177 1 physica physica PROPN cana-282 177 2 scripta scripta NOUN cana-282 177 3 93 93 NUM cana-282 177 4 , , PUNCT cana-282 177 5 no no INTJ cana-282 177 6 . . PROPN cana-282 177 7 12 12 NUM cana-282 177 8 ( ( PUNCT cana-282 177 9 2018 2018 NUM cana-282 177 10 ): ): PUNCT cana-282 177 11 125001 125001 NUM cana-282 177 12 . . PUNCT cana-282 178 1 [ [ X cana-282 178 2 10 10 NUM cana-282 178 3 ] ] SYM cana-282 178 4 bhonsale bhonsale NOUN cana-282 178 5 , , PUNCT cana-282 178 6 s. s. PROPN cana-282 178 7 s. s. PROPN cana-282 178 8 , , PUNCT cana-282 178 9 telen telen VERB cana-282 178 10 , , PUNCT cana-282 178 11 d. d. PROPN cana-282 178 12 , , PUNCT cana-282 178 13 stokbroekx stokbroekx PROPN cana-282 178 14 , , PUNCT cana-282 178 15 b. b. PROPN cana-282 178 16 , , PUNCT cana-282 178 17 & & CCONJ cana-282 178 18 van van PROPN cana-282 178 19 impe impe PROPN cana-282 178 20 , , PUNCT cana-282 178 21 j. j. PROPN cana-282 178 22 ( ( PUNCT cana-282 178 23 2019 2019 NUM cana-282 178 24 ) ) PUNCT cana-282 178 25 . . PUNCT cana-282 179 1 comparison comparison NOUN cana-282 179 2 of of ADP cana-282 179 3 numerical numerical ADJ cana-282 179 4 solution solution NOUN cana-282 179 5 strategies strategy NOUN cana-282 179 6 for for ADP cana-282 179 7 population population NOUN cana-282 179 8 balance balance NOUN cana-282 179 9 model model NOUN cana-282 179 10 of of ADP cana-282 179 11 continuous continuous ADJ cana-282 179 12 cone cone NOUN cana-282 179 13 mill mill NOUN cana-282 179 14 . . PUNCT cana-282 180 1 powder powder NOUN cana-282 180 2 technology technology NOUN cana-282 180 3 . . PUNCT cana-282 181 1 [ [ X cana-282 181 2 11 11 NUM cana-282 181 3 ] ] PUNCT cana-282 181 4 concepcion concepcion PROPN cana-282 181 5 ausin ausin PROPN cana-282 181 6 , , PUNCT cana-282 181 7 m. m. NOUN cana-282 181 8 ( ( PUNCT cana-282 181 9 2007 2007 NUM cana-282 181 10 ) ) PUNCT cana-282 181 11 an an DET cana-282 181 12 introduction introduction NOUN cana-282 181 13 to to ADP cana-282 181 14 quadrature quadrature NOUN cana-282 181 15 and and CCONJ cana-282 181 16 other other ADJ cana-282 181 17 numerical numerical ADJ cana-282 181 18 integration integration NOUN cana-282 181 19 techniques technique NOUN cana-282 181 20 , , PUNCT cana-282 181 21 encyclopedia encyclopedia NOUN cana-282 181 22 of of ADP cana-282 181 23 statistics statistic NOUN cana-282 181 24 in in ADP cana-282 181 25 quality quality NOUN cana-282 181 26 as as ADV cana-282 181 27 well well ADV cana-282 181 28 as as ADP cana-282 181 29 reliability reliability NOUN cana-282 181 30 . . PUNCT cana-282 182 1 chichester chichester PROPN cana-282 182 2 , , PUNCT cana-282 182 3 england england PROPN cana-282 182 4 . . PUNCT cana-282 183 1 [ [ X cana-282 183 2 12 12 NUM cana-282 183 3 ] ] PUNCT cana-282 183 4 rajesh rajesh PROPN cana-282 183 5 kumar kumar PROPN cana-282 183 6 sinha sinha PROPN cana-282 183 7 , , PUNCT cana-282 183 8 rakesh rakesh PROPN cana-282 183 9 kumar kumar PROPN cana-282 183 10 , , PUNCT cana-282 183 11 2010 2010 NUM cana-282 183 12 , , PUNCT cana-282 183 13 numerical numerical ADJ cana-282 183 14 method method NOUN cana-282 183 15 for for ADP cana-282 183 16 evaluating evaluate VERB cana-282 183 17 the the DET cana-282 183 18 integrable integrable ADJ cana-282 183 19 function function NOUN cana-282 183 20 on on ADP cana-282 183 21 a a DET cana-282 183 22 finite finite ADJ cana-282 183 23 interval interval NOUN cana-282 183 24 , , PUNCT cana-282 183 25 international international ADJ cana-282 183 26 journal journal NOUN cana-282 183 27 of of ADP cana-282 183 28 engineering engineering NOUN cana-282 183 29 science science NOUN cana-282 183 30 and and CCONJ cana-282 183 31 technoligy technoligy NOUN cana-282 183 32 . . PUNCT cana-282 183 33 vlo2(6 vlo2(6 NOUN cana-282 183 34 ) ) PUNCT cana-282 183 35 . . PUNCT cana-282 184 1 [ [ X cana-282 184 2 13 13 NUM cana-282 184 3 ] ] X cana-282 184 4 richard richard PROPN cana-282 184 5 lb lb PROPN cana-282 184 6 , , PUNCT cana-282 184 7 faires fair VERB cana-282 184 8 jd jd PROPN cana-282 184 9 . . PROPN cana-282 184 10 numerical numerical PROPN cana-282 184 11 analysis analysis PROPN cana-282 184 12 . . PUNCT cana-282 185 1 pacific pacific PROPN cana-282 185 2 grove grove PROPN cana-282 185 3 , , PUNCT cana-282 185 4 ca can AUX cana-282 185 5 : : PUNCT cana-282 185 6 wadsworth wadsworth PROPN cana-282 185 7 group group NOUN cana-282 185 8 ; ; PUNCT cana-282 185 9 2001 2001 NUM cana-282 185 10 . . PUNCT cana-282 186 1 [ [ X cana-282 186 2 14 14 NUM cana-282 186 3 ] ] X cana-282 186 4 jain jain PROPN cana-282 186 5 mk mk PROPN cana-282 186 6 , , PUNCT cana-282 186 7 iyengar iyengar PROPN cana-282 186 8 sr sr PROPN cana-282 186 9 , , PUNCT cana-282 186 10 jain jain PROPN cana-282 186 11 rk rk PROPN cana-282 186 12 . . PROPN cana-282 186 13 numerical numerical ADJ cana-282 186 14 methods method NOUN cana-282 186 15 for for ADP cana-282 186 16 scientific scientific ADJ cana-282 186 17 and and CCONJ cana-282 186 18 engineering engineering NOUN cana-282 186 19 computation computation NOUN cana-282 186 20 . . PUNCT cana-282 187 1 6 6 NUM cana-282 187 2 th th NUM cana-282 187 3 ed ed NOUN cana-282 187 4 . . PUNCT cana-282 188 1 new new ADJ cana-282 188 2 delhi delhi PROPN cana-282 188 3 : : PUNCT cana-282 188 4 new new ADJ cana-282 188 5 age age NOUN cana-282 188 6 international international NOUN cana-282 188 7 ; ; PUNCT cana-282 188 8 2012 2012 NUM cana-282 188 9 . . PUNCT cana-282 189 1 [ [ X cana-282 189 2 15 15 NUM cana-282 189 3 ] ] X cana-282 189 4 frontini frontini PROPN cana-282 189 5 m m PROPN cana-282 189 6 , , PUNCT cana-282 189 7 sormoni sormoni PROPN cana-282 189 8 e. e. PROPN cana-282 189 9 some some DET cana-282 189 10 variants variant NOUN cana-282 189 11 of of ADP cana-282 189 12 newton newton PROPN cana-282 189 13 ’s ’s PART cana-282 189 14 method method NOUN cana-282 189 15 with with ADP cana-282 189 16 third third ADJ cana-282 189 17 order order NOUN cana-282 189 18 convergence convergence NOUN cana-282 189 19 . . PUNCT cana-282 190 1 appl appl PROPN cana-282 190 2 math math PROPN cana-282 190 3 comput comput PROPN cana-282 190 4 2003 2003 NUM cana-282 190 5 ; ; PUNCT cana-282 190 6 140:419 140:419 NUM cana-282 190 7 - - SYM cana-282 190 8 26 26 NUM cana-282 190 9 . . PUNCT cana-282 191 1 [ [ X cana-282 191 2 16 16 NUM cana-282 191 3 ] ] X cana-282 191 4 jisheng jisheng PROPN cana-282 191 5 k k PROPN cana-282 191 6 , , PUNCT cana-282 191 7 yitian yitian PROPN cana-282 191 8 l l PROPN cana-282 191 9 , , PUNCT cana-282 191 10 xiuhua xiuhua PROPN cana-282 191 11 w. w. PROPN cana-282 191 12 third third ADJ cana-282 191 13 - - PUNCT cana-282 191 14 order order NOUN cana-282 191 15 modification modification NOUN cana-282 191 16 of of ADP cana-282 191 17 newton newton PROPN cana-282 191 18 ’s ’s PART cana-282 191 19 method method NOUN cana-282 191 20 . . PUNCT cana-282 192 1 j j PROPN cana-282 192 2 comput comput PROPN cana-282 192 3 appl appl PROPN cana-282 192 4 math math PROPN cana-282 192 5 2007 2007 NUM cana-282 192 6 ; ; PUNCT cana-282 192 7 205:1 205:1 NUM cana-282 192 8 - - SYM cana-282 192 9 5 5 NUM cana-282 192 10 . . PUNCT cana-282 193 1 [ [ X cana-282 193 2 17 17 NUM cana-282 193 3 ] ] X cana-282 193 4 jayakumar jayakumar PROPN cana-282 193 5 j j PROPN cana-282 193 6 , , PUNCT cana-282 193 7 kalyanasundaram kalyanasundaram PROPN cana-282 193 8 m. m. PROPN cana-282 193 9 modified modified PROPN cana-282 193 10 newton newton PROPN cana-282 193 11 ’s ’s PART cana-282 193 12 method method NOUN cana-282 193 13 using use VERB cana-282 193 14 harmonic harmonic ADJ cana-282 193 15 mean mean NOUN cana-282 193 16 for for ADP cana-282 193 17 solving solve VERB cana-282 193 18 nonlinear nonlinear ADJ cana-282 193 19 equations equation NOUN cana-282 193 20 . . PUNCT cana-282 194 1 iosr iosr PROPN cana-282 194 2 j j PROPN cana-282 194 3 math math PROPN cana-282 194 4 2013 2013 NUM cana-282 194 5 ; ; PUNCT cana-282 194 6 7:93 7:93 NUM cana-282 194 7 - - SYM cana-282 194 8 7 7 NUM cana-282 194 9 . . PUNCT cana-282 195 1 [ [ X cana-282 195 2 18 18 NUM cana-282 195 3 ] ] X cana-282 195 4 muthumalai muthumalai PROPN cana-282 195 5 r r PROPN cana-282 195 6 k. k. PROPN cana-282 195 7 some some DET cana-282 195 8 formulae formulae NOUN cana-282 195 9 for for ADP cana-282 195 10 numerical numerical ADJ cana-282 195 11 differentiation differentiation NOUN cana-282 195 12 through through ADP cana-282 195 13 divided divide VERB cana-282 195 14 difference difference NOUN cana-282 195 15 . . PUNCT cana-282 196 1 international international ADJ cana-282 196 2 journal journal PROPN cana-282 196 3 of of ADP cana-282 196 4 mathematical mathematical ADJ cana-282 196 5 archive archive NOUN cana-282 196 6 . . PUNCT cana-282 197 1 2012 2012 NUM cana-282 197 2 , , PUNCT cana-282 197 3 03 03 NUM cana-282 197 4 . . PUNCT cana-282 198 1 [ [ X cana-282 198 2 19 19 NUM cana-282 198 3 ] ] X cana-282 198 4 khan khan PROPN cana-282 198 5 m m PROPN cana-282 198 6 m m PROPN cana-282 198 7 , , PUNCT cana-282 199 1 hossain hossain PROPN cana-282 199 2 m m PROPN cana-282 199 3 r r PROPN cana-282 199 4 and and CCONJ cana-282 199 5 selina selina PROPN cana-282 199 6 parvin parvin PROPN cana-282 199 7 . . PUNCT cana-282 200 1 numerical numerical ADJ cana-282 200 2 integration integration NOUN cana-282 200 3 schemes scheme NOUN cana-282 200 4 for for ADP cana-282 200 5 unequal unequal ADJ cana-282 200 6 data datum NOUN cana-282 200 7 spacing space VERB cana-282 200 8 . . PUNCT cana-282 201 1 american american ADJ cana-282 201 2 journal journal PROPN cana-282 201 3 of of ADP cana-282 201 4 applied applied ADJ cana-282 201 5 mathematics mathematic NOUN cana-282 201 6 . . PUNCT cana-282 202 1 2017 2017 NUM cana-282 202 2 , , PUNCT cana-282 202 3 5(2 5(2 NUM cana-282 202 4 ) ) PUNCT cana-282 202 5 , , PUNCT cana-282 202 6 48 48 NUM cana-282 202 7 - - SYM cana-282 202 8 56 56 NUM cana-282 202 9 . . PUNCT cana-282 203 1 [ [ X cana-282 203 2 20 20 NUM cana-282 203 3 ] ] X cana-282 203 4 thukral thukral PROPN cana-282 203 5 r. r. PROPN cana-282 203 6 further further ADJ cana-282 203 7 acceleration acceleration NOUN cana-282 203 8 of of ADP cana-282 203 9 the the DET cana-282 203 10 simpson simpson PROPN cana-282 203 11 ’s ’s PART cana-282 203 12 method method NOUN cana-282 203 13 for for ADP cana-282 203 14 solving solve VERB cana-282 203 15 non non ADJ cana-282 203 16 - - ADJ cana-282 203 17 linear linear ADJ cana-282 203 18 equation equation NOUN cana-282 203 19 . . PUNCT cana-282 204 1 journal journal NOUN cana-282 204 2 of of ADP cana-282 204 3 advances advance NOUN cana-282 204 4 in in ADP cana-282 204 5 mathematics mathematic NOUN cana-282 204 6 . . PUNCT cana-282 205 1 2018 2018 NUM cana-282 205 2 , , PUNCT cana-282 205 3 14 14 NUM cana-282 205 4 , , PUNCT cana-282 205 5 02 02 NUM cana-282 205 6 . . PUNCT cana-282 206 1 [ [ X cana-282 206 2 21 21 NUM cana-282 206 3 ] ] X cana-282 206 4 jayakumar jayakumar PROPN cana-282 206 5 j. j. PROPN cana-282 206 6 generalized generalized PROPN cana-282 206 7 simpson simpson PROPN cana-282 206 8 - - PUNCT cana-282 206 9 newton newton PROPN cana-282 206 10 's 's PART cana-282 206 11 method method NOUN cana-282 206 12 for for ADP cana-282 206 13 solving solve VERB cana-282 206 14 nonlinear nonlinear ADJ cana-282 206 15 equations equation NOUN cana-282 206 16 with with ADP cana-282 206 17 cubic cubic ADJ cana-282 206 18 convergence convergence NOUN cana-282 206 19 . . PUNCT cana-282 207 1 iosr iosr ADJ cana-282 207 2 journal journal PROPN cana-282 207 3 of of ADP cana-282 207 4 mathematics mathematic NOUN cana-282 207 5 . . PUNCT cana-282 208 1 2013 2013 NUM cana-282 208 2 , , PUNCT cana-282 208 3 7(5 7(5 NUM cana-282 208 4 ) ) PUNCT cana-282 208 5 , , PUNCT cana-282 208 6 58 58 NUM cana-282 208 7 - - SYM cana-282 208 8 61 61 NUM cana-282 208 9 . . PUNCT cana-282 209 1 [ [ X cana-282 209 2 22 22 NUM cana-282 209 3 ] ] X cana-282 209 4 eskandari eskandari PROPN cana-282 209 5 h. h. PROPN cana-282 209 6 simpson simpson PROPN cana-282 209 7 ’s ’s PART cana-282 209 8 method method NOUN cana-282 209 9 solution solution NOUN cana-282 209 10 for for ADP cana-282 209 11 non non ADJ cana-282 209 12 - - ADJ cana-282 209 13 linear linear ADJ cana-282 209 14 equation equation NOUN cana-282 209 15 , , PUNCT cana-282 209 16 scientific scientific ADJ cana-282 209 17 research research NOUN cana-282 209 18 publishing publishing NOUN cana-282 209 19 inc.2017 inc.2017 ADP cana-282 209 20 , , PUNCT cana-282 209 21 8 8 NUM cana-282 209 22 , , PUNCT cana-282 209 23 929 929 NUM cana-282 209 24 - - SYM cana-282 209 25 933 933 NUM cana-282 209 26 . . PUNCT cana-282 210 1 [ [ X cana-282 210 2 23 23 NUM cana-282 210 3 ] ] X cana-282 210 4 chapra chapra PROPN cana-282 210 5 , , PUNCT cana-282 210 6 steven steven PROPN cana-282 210 7 c. c. PROPN cana-282 210 8 , , PUNCT cana-282 210 9 and and CCONJ cana-282 210 10 raymond raymond PROPN cana-282 210 11 p. p. PROPN cana-282 210 12 canale canale PROPN cana-282 210 13 . . PUNCT cana-282 211 1 numerical numerical ADJ cana-282 211 2 methods method NOUN cana-282 211 3 for for ADP cana-282 211 4 engineers engineer NOUN cana-282 211 5 . . PUNCT cana-282 212 1 7th 7th ADJ cana-282 212 2 ed ed NOUN cana-282 212 3 . . PUNCT cana-282 212 4 ,pp ,pp PROPN cana-282 212 5 . . PUNCT cana-282 213 1 605615 605615 NUM cana-282 213 2 , , PUNCT cana-282 213 3 ( ( PUNCT cana-282 213 4 2015 2015 NUM cana-282 213 5 ) ) PUNCT cana-282 213 6 . . PUNCT cana-282 214 1 [ [ X cana-282 214 2 24 24 NUM cana-282 214 3 ] ] SYM cana-282 214 4 fornberg fornberg PROPN cana-282 214 5 , , PUNCT cana-282 214 6 bengt bengt NOUN cana-282 214 7 . . PUNCT cana-282 215 1 " " PUNCT cana-282 215 2 improving improve VERB cana-282 215 3 the the DET cana-282 215 4 accuracy accuracy NOUN cana-282 215 5 of of ADP cana-282 215 6 the the DET cana-282 215 7 trapezoidal trapezoidal ADJ cana-282 215 8 rule rule NOUN cana-282 215 9 . . PUNCT cana-282 215 10 " " PUNCT cana-282 216 1 siam siam PROPN cana-282 216 2 review review NOUN cana-282 216 3 63.1 63.1 NUM cana-282 216 4 ( ( PUNCT cana-282 216 5 2021 2021 NUM cana-282 216 6 ): ): PUNCT cana-282 216 7 167 167 NUM cana-282 216 8 - - SYM cana-282 216 9 180 180 NUM cana-282 216 10 . . PUNCT cana-282 217 1 [ [ X cana-282 217 2 25 25 NUM cana-282 217 3 ] ] PUNCT cana-282 217 4 trefethen trefethen NOUN cana-282 217 5 , , PUNCT cana-282 217 6 lloyd lloyd PROPN cana-282 217 7 n. n. PROPN cana-282 217 8 , , PUNCT cana-282 217 9 and and CCONJ cana-282 217 10 j. j. PROPN cana-282 217 11 a. a. PROPN cana-282 217 12 c. c. PROPN cana-282 217 13 weideman weideman NOUN cana-282 217 14 . . PUNCT cana-282 218 1 " " PUNCT cana-282 218 2 the the DET cana-282 218 3 exponentially exponentially ADV cana-282 218 4 convergent convergent ADJ cana-282 218 5 trapezoidal trapezoidal ADJ cana-282 218 6 rule rule NOUN cana-282 218 7 . . PUNCT cana-282 218 8 " " PUNCT cana-282 219 1 siam siam PROPN cana-282 219 2 review review NOUN cana-282 219 3 56.3 56.3 NUM cana-282 219 4 ( ( PUNCT cana-282 219 5 2014 2014 NUM cana-282 219 6 ): ): PUNCT cana-282 219 7 385 385 NUM cana-282 219 8 - - SYM cana-282 219 9 458 458 NUM cana-282 219 10 . . PUNCT cana-282 220 1 [ [ X cana-282 220 2 26 26 NUM cana-282 220 3 ] ] X cana-282 220 4 bailey bailey NOUN cana-282 220 5 , , PUNCT cana-282 220 6 david david PROPN cana-282 220 7 h. h. PROPN cana-282 220 8 , , PUNCT cana-282 220 9 and and CCONJ cana-282 220 10 jonathan jonathan PROPN cana-282 220 11 m. m. PROPN cana-282 220 12 borwein borwein PROPN cana-282 220 13 . . PUNCT cana-282 221 1 " " PUNCT cana-282 221 2 high high ADJ cana-282 221 3 - - PUNCT cana-282 221 4 precision precision NOUN cana-282 221 5 numerical numerical ADJ cana-282 221 6 integration integration NOUN cana-282 221 7 : : PUNCT cana-282 221 8 progress progress NOUN cana-282 221 9 and and CCONJ cana-282 221 10 challenges challenge NOUN cana-282 221 11 . . PUNCT cana-282 221 12 " " PUNCT cana-282 222 1 journal journal NOUN cana-282 222 2 of of ADP cana-282 222 3 symbolic symbolic ADJ cana-282 222 4 computation computation NOUN cana-282 222 5 46.7 46.7 NUM cana-282 222 6 ( ( PUNCT cana-282 222 7 2011 2011 NUM cana-282 222 8 ): ): PUNCT cana-282 222 9 741 741 NUM cana-282 222 10 - - SYM cana-282 222 11 754 754 NUM cana-282 222 12 . . PUNCT