Journal of Multidisciplinary Applied Natural Science ACCEPTED MANUSCRIPT • OPEN ACCESS Integration of Rational Functions To cite this article before publication: L. Rathour, D. Obradovic, K. Khatri, S. K. Tiwari, L. N. Mishra, and V. N. Mishra. (2023). J. Multidiscip. Appl. Nat. Sci. in press. https://doi.org/10.47352/jmans.2774-3047.186. Manuscript version: Accepted Manuscript Accepted Manuscript is “the version of the article accepted for publication including all changes made as a result of the peer review process, and which may also include the addition to the article by Pandawa Institute of a header, an article ID, a cover sheet and/or an ‘Accepted Manuscript’ watermark, but excluding any other editing, typesetting or other changes made by Pandawa Institute and/or its licensors” This Accepted Manuscript is © 2023 The Author(s). 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View the article online for updates and enhancements. https://doi.org/10.47352/jmans.2774-3047.186 https://creativecommons.org/licenses/by/4.0/ https://doi.org/10.47352/jmans.2774-3047.186 Journal of Multidisciplinary Applied Natural Science Integration of Rational Functions 1 2 ; 4,d*)Tiwari Kant Shiv ;3,c)Khatri Kejal ;2,b)Obradovic Dragan ;1,a)Rathour Laxmi3 6,f); Vishnu Narayan Mishra 5,e)Lakshmi Narayan Mishra 4 5 (India) 796012- Technology, MizoramDepartment of Mathematics, National Institute of 1 6 )Serbia( 12208-Elementary School "Jovan Cvijic", Pozarevac2 7 )India( 314403- Dungarpur Simalwara, College, Government Mathematics, of Department3 8 )India(363642 -Department of Mathematics, Lukhdhirji Engineering College, Morbi4 9 dia)In(632014 -Vellore of Technology, Institute Vellore Mathematics, of Department5 10 )India(484887 -Amarkantak University, Tribal Gandhi National Mathematics, Indira of Department6 11 laxmirathour817@gmail.com a) 12 dragishaobradovic@yahoo.comb) 13 kejal0909@gmail.comc) 14 shivkant.math@gmail.com :Correspondence d) 15 lakshminarayanmishra04@gmail.come) 16 vishnunarayanmishra@gmail.comf) 17 18 ORCIDs: 19 First Author : https://orcid.org/0000-0002-2659-7568 20 Second Author : https://orcid.org/0000-0001-5871-6958 21 Third Author : https://orcid.org/0000-0002-3425-1727 22 Fourth Author : https://orcid.org/0000-0003-0942-3467 23 Fifth Author : https://orcid.org/0000-0001-7774-7290 24 Sixth Author : https://orcid.org/0000-0002-2159-7710 25 26 27 AUTHOR CONTRIBUTIONS 28 29 All authors equally contributed to this paper. 30 31 CONFLICT OF INTEREST 32 33 The authors declare that there is no conflict of interest. 34 35 36 37 38 39 40 41 42 43 44 45 46 47 A CC EP TE D M A N U SC RI PT mailto:laxmirathour817@gmail.com mailto:b)dragishaobradovic@yahoo.com mailto:c)kejal0909@gmail.com mailto:shivkant.math@gmail.com mailto:e)lakshminarayanmishra04@gmail.com mailto:f)vishnunarayanmishra@gmail.com https://orcid.org/0000-0002-2659-7568 https://orcid.org/0000-0001-5871-6958 https://orcid.org/0000-0002-3425-1727 https://orcid.org/0000-0003-0942-3467 https://orcid.org/0000-0001-7774-7290 Journal of Multidisciplinary Applied Natural Science Integration of Rational Functions 1 2 Abstract. A rational function can always be integrated, that is, the integral of such a function 3 is always an elementary function. The integration procedure is complex and consists of four 4 steps: elimination of the common zero-points of the numerator and denominator, reduction to 5 a true rational function, decomposition into partial fractions and integration of the obtained 6 expressions using direct integration, substitution method or partial integration method. 7 Integrating rational functions is important because integrals of rational functions of 8 trigonometric functions as well as integrals of some irrational functions are reduced to 9 integrals of rational functions by appropriate transformations. 10 11 Key words: mathematics; rational functions; polynomial functions; partial fractions. 12 13 1. INTRODUCTION 14 15 The tasks of teaching higher mathematics are to demonstrate to students the operation of 16 the laws of materialistic dialectics, the essence of the scientific approach, the specifics of 17 mathematics and its role in the realization of scientific and technological progress, using 18 examples of mathematical concepts and methods. It is necessary to teach students the 19 methods of research and solving mathematical formalized tasks, to develop in students the 20 ability to analyze the obtained results, to inculcate in them the skills of independent study of 21 literature on mathematics and its application. 22 Every rational fraction is integrated into elementary functions, but studying the general 23 integration algorithm and alternative approaches is quite a difficult task. From lectures and 24 seminars, it is impossible to sufficiently consider all the details, then the main part of the 25 material is usually discussed in the classroom, and the rest is given to students for 26 independent study. This paper was created within the educational and methodological 27 complex of the discipline "Mathematical Analysis" and is dedicated to the topic "Integration 28 of Rational Fractions". It considers those cases and methods that cannot be considered in their 29 entirety in the classroom. 30 Integration of rational fractions. Expression of the form 𝑃𝑚(𝑥) 𝑄𝑛(𝑥) , where Pm(x) and Qn(x) 31 polynomials m-th and n-th degree, is called a rational fraction. A rational fraction is called 32 A CC EP TE D M A N U SC RI PT Journal of Multidisciplinary Applied Natural Science proper if mn and improper if m  n . The problem of integrating a rational fraction can be 1 reduced to the problem integration of a proper rational fraction, since any improper rational 2 fraction can be represented by division by a column in as the sum of a polynomial and a 3 proper rational fraction. We recite some research papers and books [1]-[7] for the right 4 direction of integration of rational functions and its applications. 5 6 2. REPRESENTATION OF A RATIONAL FUNCTIONS AS A SUM OF 7 POLYNOMIALS 8 9 Those who say that trigonometry is not needed in real life are not needed on the road. So, 10 what are its common applied tasks? Measure the distance between inaccessible objects. The 11 triangle technique is of great importance, which enables the measurement of distances to 12 nearby stars in astronomy, between benchmarks in geography, control satellite navigation 13 systems. The use of trigonometric techniques should also be noted, such as navigation 14 techniques, music theory, acoustics, optics, financial market analysis, electronics, probability 15 theory, statistics, biology, medicine (ultrasound and computed tomography), pharmacology, 16 chemistry, number theory (and, as a result, cryptology, seismology, meteorology, 17 oceanology, cartography, many parts of physics, topography and geodesy, architecture, 18 phonetics, economics, electronic equipment, mechanical engineering, computer graphics, 19 crystallography etc. 20 A rational function is a ratio of polynomials where the polynomial in the denominator 21 should not be equal to zero. Doesn't that look like the definition of a rational number (which 22 is of the form p/k, where k = 0)? 23 Polynomial functions play an important role in mathematics. They are generally simple to 24 compute (requiring only calculations that can be performed manually) and can be used to 25 model many real-world phenomena. In fact, scientists and mathematicians often simplify 26 complex mathematical models by substituting a polynomial model that is "close enough" for 27 their purposes. 28 Let it be 𝑓(𝑥) = 𝑃𝑛(𝑥) 𝑄𝑚(𝑥) rational function. If , it is 𝑛 ≥ 𝑚 , i.e. not a real rational function, 29 then by division 𝑃𝑛 (𝑥) with 𝑄𝑚 (𝑥), 𝑓 can be represented as a sum of polynomials and real 30 rational functions in the form 31 𝑓(𝑥) = 𝑃(𝑥) + 𝑅(𝑥) 𝑄𝑚 (𝑥) 32 A CC EP TE D M A N U SC RI PT Journal of Multidisciplinary Applied Natural Science Let's say it is 𝑓(𝑥) = 𝑃𝑛(𝑥) 𝑄𝑚(𝑥) true rational function (𝑛 < 𝑚). Then it can be decomposed into 1 a sum of partial fractions of the form 2 3 𝐴 (𝑥−𝑎)𝑘 that is 𝐴𝑥+𝐵 (𝑥2+𝑏𝑥+𝑐)𝑘 4 5 where the first type of fraction comes from real, and the second from the type of complex 6 zero-points of polynomials in the denominator of the function f. Therefore, the problem is 7 reduced to solving the integral of the form 8 9 ∫ 𝐴 (𝑥−𝑎)𝑘 𝑑𝑥 and ∫ 𝐴𝑥+𝐵 (𝑥2+𝑏𝑥+𝑐)𝑘 𝑑𝑥 10 11 where A and B are some real constants, is a natural number, and is a quadratic trinomial 12 𝑥2 + 𝑏𝑥 + 𝑐 has complex-conjugate zero-points. 13 14 Integrals of the form ∫ 𝐴 (𝑥−𝑎)𝑘 𝑑𝑥 15 1. If it is 𝑘 = 1, he gives ∫ 𝐴 (𝑥−𝑎) 𝑑𝑥 = 𝐴 ln|𝑥 − 𝑎| + 𝐶. 16 2. If it is 𝑘 > 1, then by changing the variable 𝑥 − 𝑎 = 𝑡 ⟹ 𝑑𝑥 = 𝑑𝑡, we have it 17 ∫ 𝐴 (𝑥−𝑎)𝑘 𝑑𝑥 = { 𝑥 − 𝑎 = 𝑡 𝑑𝑥 = 𝑑𝑡 } = 𝐴 ∫ 𝑡 −𝑘 𝑑𝑡 = 𝐴 𝑡−𝑘+1 −𝑘+1 + 𝐶 = 𝐴 (1−𝑘)(𝑥−𝑎)1−𝑘 + 𝐶. 18 19 3. REDUCTION TO TRUE RATIONAL FUNCTION 20 21 A rational function is a true rational function if the degree of the numerator is less than the 22 degree of the denominator. If the degree of the numerator is greater than or equal to the 23 degree of the denominator, we can divide the polynomials, so we have 24 25 𝑓(𝑥) = 𝑝(𝑥) 𝑞(𝑥) = 𝑠(𝑥) + 𝑟(𝑥) 𝑞(𝑥) 26 27 where r and s are polynomials. The polynomial is the remainder when dividing, and the 28 degree of r is ᶴ s (x) dx less than the power of q. Of course, it is solved by direct integration, 29 so we can conclude the following: 30 A CC EP TE D M A N U SC RI PT Journal of Multidisciplinary Applied Natural Science Integrating rational functions is reduced to integrating real rational functions of the form 1 2 𝑓(𝑥) = 𝑝(𝑥) 𝑞(𝑥) 3 4 where p and q have no common zero-points and the degree of p is less than the degree of 5 q. 6 Example: Dividing a polynomial give 7 8 𝑥3+𝑥 𝑥2−1 = 𝑥 + 2𝑥 𝑥2−1 , 9 10 so, it is ∫ 𝑥3+𝑥 𝑥2−1 𝑑𝑥 = ∫ (𝑥 + 2𝑥 𝑥2−1 ) 𝑑𝑥 11 = ∫ 𝑥 𝑑𝑥 + ∫ 2𝑥 𝑥2−1 𝑑𝑥 = 𝑥2 2 + ∫ 𝑑𝑡 𝑡 , putting { 𝑥2 − 1 = 𝑡, 2𝑥 𝑑𝑥 = 𝑑𝑡 } 12 = 𝑥2 2 + ln|𝑡| + 𝐶 = 𝑥2 2 + ln|𝑥2 − 1| + 𝐶. 13 14 4. INTEGRATION BY PARTIAL FRACTIONS 15 16 Integration by partial fractions is a method used to decompose and then integrate a rational 17 fraction that has complex terms in the denominator. By using a partial fraction, we calculate 18 and decompose the expression into simpler terms so that we can easily calculate or integrate 19 the resulting expression. 20 The basic idea in integration by partial fractions is to factor the denominator and then 21 factor them into two different fractions where the denominators are the factors and the 22 numerator is calculated accordingly. Let's learn more about the different forms used in 23 integration by partial fractions, as well as the different methods. 24 25 4.1. We decompose the sub integral function into partial fractions 26 27 𝑥−3 𝑥3−𝑥 = 𝑥−3 𝑥(𝑥2−1) = 𝑥−3 𝑥(𝑥−1)(𝑥+1) = 𝐴 𝑥 + 𝐵 𝑥−1 + 𝐶 𝑥+1 28 29 Using the polynomial similarity method, we get that 𝐴 = 3, 𝐵 = −1, 𝐶 = −2 30 31 A CC EP TE D M A N U SC RI PT Journal of Multidisciplinary Applied Natural Science ∫ 𝑥−3 𝑥3−𝑥 𝑑𝑥 = ∫ 𝑥−3 𝑥(𝑥−1)(𝑥+1) 𝑑𝑥 = ∫ [ 3 𝑥 − 1 𝑥−1 − 2 𝑥+1 ] 𝑑𝑥 1 = ∫ 3 𝑥 𝑑𝑥 − ∫ 1 𝑥−1 𝑑𝑥 − ∫ 2 𝑥+1 𝑑𝑥 = 3 ln|𝑥| − ln|𝑥 − 1| − 2ln|𝑥 + 1| + 𝐶 2 = ln | 𝑥3 (𝑥−1)(𝑥+1)2 | + 𝐶 3 4 Integrals of the form ∫ 𝐴𝑥+𝐵 (𝑥2+𝑏𝑥+𝑐) 𝑑𝑥. We have already solved integrals of this form using 5 the shift method 6 7 ∫ 𝐴𝑥+𝐵 (𝑥2+𝑏𝑥+𝑐) 𝑑𝑥 = ∫ 𝐴(2𝑥+𝑏)+(𝐵− 𝐴𝑏 2 ) 𝑎𝑥2+𝑏𝑥+𝑐 𝑑𝑥 8 = 𝐴 2 ∫ 2𝑥+𝑏 𝑎𝑥2+𝑏𝑥+𝑐 𝑑𝑥 + (𝐵 − 𝐴𝑏 2 ) . ∫ 𝑑𝑥 𝑎𝑥2+𝑏𝑥+𝑐 . 9 10 Where in ∫ 2𝑎𝑥+𝑏 𝑥2+𝑏𝑥+𝑐 𝑑𝑥 = ∫ (𝑎𝑥2+𝑏𝑥+𝑐) 𝑎𝑥2+𝑏𝑥+𝑐 𝑑𝑥 = ln|𝑎𝑥2 + 𝑏𝑥 + 𝑐| + 𝐶 and 11 12 ∫ 𝑑𝑥 𝑥2+𝑏𝑥+𝑐 = ∫ 𝑑𝑥 (𝑥+ 𝑏 2 )2+(𝑐− 𝑏2 42 ) putting { 𝑥 + 𝑏 2 = 𝑡, 𝑐 − 𝑏2 42 = 𝑘2 𝑑𝑥 = 𝑑𝑡 } 13 = ∫ 𝑑𝑡 𝑡2+𝑘 2 = 1 𝑘 𝑎𝑟𝑐𝑡𝑔𝑥 𝑡 𝑘 + 𝐶 = 1 √𝑐− 𝑏2 42 𝑎𝑟𝑐𝑡𝑔𝑥 𝑥+ 𝑏 2 √𝑐− 𝑏2 42 + 𝐶. 14 15 Integrals of the form ∫ 𝐴𝑥+𝐵 (𝑥2+𝑏𝑥+𝑐)𝑘 𝑑𝑥, wherein 𝑘 ≥ 2, 𝑏2 − 4𝑐 < 0. Let's perform the 16 transformation of the sub integral function 17 18 ∫ 𝐴𝑥+𝐵 (𝑥2+𝑏𝑥+𝑐)𝑘 𝑑𝑥 = ∫ 𝐴(2𝑥+𝑏)+(𝐵− 𝐴𝑏 2 ) (𝑥2+𝑏𝑥+𝑐)𝑘 𝑑𝑥 19 = 𝐴 2 ∫ 2𝑥+𝑏 (𝑥2+𝑏𝑥+𝑐)𝑘 𝑑𝑥 + (𝐵 − 𝐴𝑏 2 ) . ∫ 𝑑𝑥 (𝑥2+𝑏𝑥+𝑐)𝑘 . 20 We can solve the first integral by shifting, where's he from 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 𝑡. So, we get 21 (2𝑥 + 𝑏)𝑑𝑥 = 𝑑𝑡. 22 23 𝐼0 = ∫ 𝐴𝑥+𝐵 (𝑥2+𝑏𝑥+𝑐)𝑘 𝑑𝑥 = ∫ 𝑑𝑡 𝑡𝑘 = 1 (1−𝑘)𝑡 𝑘−1 + 𝐶 = 1 (1−𝑘)(𝑥2+𝑏𝑥+𝑐)𝑘−1 + 𝐶. 24 25 Let's denote the second integral by 𝐼𝑘 = ∫ 𝑑𝑥 (𝑎𝑥2+𝑏𝑥+𝑐)𝑘 and perform the transformation 26 A CC EP TE D M A N U SC RI PT Journal of Multidisciplinary Applied Natural Science 1 𝐼𝑘 = ∫ 𝑑𝑥 (𝑎𝑥2+𝑏𝑥+𝑐)𝑘 = ∫ 𝑑𝑥 [(𝑥+ 𝑏 2 ) 2 +(𝑐− 𝑏2 42 )] 𝑘 2 3 Introducing a shift 𝑥 + 𝑏 2 = 𝑡, 𝑑𝑥 = 𝑑𝑡 and label 𝑐 − 𝑏2 42 = 𝑙2 and 𝐼𝑘 we get 4 5 𝐼𝑘 = ∫ 𝑑𝑥 𝑡2+𝑙2 . 6 7 By combining the variable change method and partial integration, after a series of 8 relations, a recursive formula for calculating the integral is obtained, which we will not derive 9 here, but give it in its finished form 10 11 𝐼𝑘 = 𝑡 2𝑙(𝑘−1)(𝑡2+𝑙2)𝑘−1 + 2𝑘−3 2𝑙2(𝑘−1) ∙ ∫ 𝑑𝑡 (𝑡 2+𝑙2)𝑘−1 , 𝐼1 = ∫ 𝑑𝑡 𝑡2+𝑙2 = 1 𝑙 𝑎𝑟𝑐𝑡𝑔𝑥 𝑡 𝑙 + 𝐶 12 𝐼𝑘−1 13 14 Example: Let's calculate the integral ∫ 𝑥−1 (𝑥2+2𝑥+3)2 15 How it is 𝑝 = 2 and 𝑞 = 3, it is 𝑝2 − 4𝑞 = −8 < 0, so the denominator has no real zero-16 points. Let's transform the integral 17 𝐼 = ∫ 𝑥−1 (𝑥2+2𝑥+3𝑐)2 𝑑𝑥 = ∫ 1 2 (2𝑥+2)+(−1−1) (𝑥2+2𝑥+2)2 𝑑𝑥 = 1 2 ∫ 2𝑥+2 (𝑥2+2𝑥+3)2 𝑑𝑥 − 2 ∙ ∫ 𝑑𝑥 (𝑥2+2𝑥+3𝑐)2 . 18 19 𝐼0 = ∫ 2𝑥+2 (𝑥2+2𝑥+3𝑐)2 𝑑𝑥 = { 𝑥2 + 2𝑥 + 3𝑐 = 𝑡 (𝑥2 + 2𝑥 + 3𝑐)𝑑𝑥 = (2𝑥 + 2)𝑑𝑥 = 𝑑𝑡 } 20 = ∫ 𝑑𝑡 𝑡2 = − 1 𝑡 + 𝐶 = 1 𝑥2+2𝑥+3𝑐 + 𝐶. 21 𝐼2 = ∫ 𝑑𝑥 (𝑎𝑥2+𝑏𝑥+𝑐)2 = ∫ 𝑑𝑥 [(𝑥+1)2+(3−1)]2 = { 𝑥 + 1 = 𝑡 𝑑𝑥 = 𝑑𝑡 } = 𝑑𝑡 [𝑡2+2]2 . 22 23 By substituting into the recurrent formula, we get 24 𝐼2 = 𝑡 2∙2(2−1)(𝑡2+2)2−1 + 2∙2−3 2∙2(2−1) ∙ ∫ 𝑑𝑡 (𝑡2+2)2−1 , where 𝐼1 = ∫ 𝑑𝑡 𝑡2 +2 = 1 √2 𝑎𝑟𝑐𝑡𝑔 𝑡 2 + 𝐶, 25 𝐼2−1=1 26 27 so, it is 𝐼2 = 𝑡 4(𝑡2+2) + 1 4 ∙ 1 √2 𝑎𝑟𝑐𝑡𝑔 𝑡 √2 + 𝐶 = {𝑡 = 𝑥 + 1} 28 A CC EP TE D M A N U SC RI PT Journal of Multidisciplinary Applied Natural Science = 𝑥+1 4(𝑥2+2𝑥+3) + 1 4√2 𝑎𝑟𝑐𝑡𝑔 𝑥+1 √2 + 𝐶. 1 2 So, it is the final solution 3 𝐼 = ∫ 𝑥−1 (𝑥2+2𝑥+3𝑐)2 𝑑𝑥 = 1 2 𝐼0 − 2𝐼2 4 = 1 𝑥2+2𝑥+3𝑐 + 𝑥+1 4(𝑥2+2𝑥+3) + 1 4√2 𝑎𝑟𝑐𝑡𝑔 𝑥+1 √2 + 𝐶 5 = 𝑥+2 4(𝑥2+2𝑥+3) + 1 4√2 𝑎𝑟𝑐𝑡𝑔 𝑥+1 √2 + 𝐶. 6 7 5. CONTRIBUTION TO THE INTEGRATION OF RATIONAL 8 FUNCTIONS 9 10 Mathematical thinking and reasoning are the main goal of mathematics education. It is 11 known from research that tasks involving problem solving, modelling and argumentation 12 provide excellent opportunities to realize students' mathematical thinking [8]. Many topics in 13 secondary education are already taught through tasks of this type (e.g. elementary algebra, 14 exponential functions). However, there are still topics that are traditionally taught formally 15 and mechanically, which makes it difficult for students to think and communicate in a 16 conceptual sense. This is the case for the integral concept, which was introduced in most 17 European countries in the last two years of secondary education and taught in a procedural 18 way. 19 The integration of rational functions involves a number of cases that require different 20 procedures. These procedures are based on operations such as calculating derivatives, 21 dividing polynomials, factoring, and solving equations; everything is applied in order to 22 decompose to the original integral in simpler cases that can be approached using already 23 known techniques. 24 To find the rational part, we first need to know about square-free factorization. An 25 important result in algebra is that every polynomial with rational coefficients can be uniquely 26 factored into irreducible polynomials with rational coefficients, up to multiplication by a 27 nonzero constant and rearrangement of factors, just as every integer can be uniquely factored 28 into prime up to multiples of 1 and -1 and redistribution of factors (technically, it is with 29 coefficients from the unique domain of factorization, for which rationales are a special case, 30 up to the multiplication of unity, which for rationales is every constant except zero). 31 A CC EP TE D M A N U SC RI PT Journal of Multidisciplinary Applied Natural Science To integrate the corresponding rational function, we can apply the method of partial 1 fractions. This method allows converting the integral of a complicated rational function into a 2 sum of integrals of simpler functions. The denominators of partial fractions can contain 3 unrepeated linear factors, repeated linear factors, unrepeated irreducible quadratic factors, 4 and repeated irreducible quadratic factors. 5 6 6. CONCLUSION 7 8 The integration of rational functions includes a variety of cases requiring different 9 procedures. These procedures are based on operations such as derivative calculation, division 10 of polynomials, factorization and resolution of equations; all applied in order to decompose to 11 the original integral in simpler cases that can be approached using the techniques already 12 known. 13 14 REFERENCES 15 16 [1] C. Canuto and A. Tabacco. (2015). "Mathematical Analysis I". Springer Cham, 17 Switzerland. 10.1007/978-3-319-12772-9. 18 [2] D. Lazard and R. Rioboo. (1990). "Integration of rational functions: Rational 19 computation of the logarithmic part". Journal of Symbolic Computation. 9 (2): 113-115. 20 10.1016/s0747-7171(08)80026-0. 21 [3] R. H. C. Moir, R. M. Corless, M. M. 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"Algebraic factoring and rational function integration". 31 Proceedings of the Third ACM Symposium on Symbolic and Algebraic Computation. 32 10.1145/800205.806338. 33 A CC EP TE D M A N U SC RI PT https://doi.org/10.1007/978-3-319-12772-9 https://doi.org/10.1016/s0747-7171(08)80026-0 https://doi.org/10.1007/s11075-019-00726-6 https://doi.org/10.1016/0377-0427(92)90182-w https://doi.org/10.1145/96877.96973 https://doi.org/10.1145/800205.806338 Journal of Multidisciplinary Applied Natural Science [7] K. Gürlebeck, K. Habetha, and W. Sprößig. (2007). "Holomorphic functions in the 1 plane and n-dimensional space". Springer Science & Business Media, Basel. 2 [8] J. Lithner. (2007). "A research framework for creative and imitative reasoning". 3 Educational Studies in Mathematics. 67 (3): 255-276. 10.1007/s10649-007-9104-2. 4 5 A CC EP TE D M A N U SC RI PT https://doi.org/10.1007/s10649-007-9104-2 1. INTRODUCTION 3. REDUCTION TO TRUE RATIONAL FUNCTION 4. INTEGRATION BY PARTIAL FRACTIONS 6. CONCLUSION