id	sid	tid	token	lemma	pos
ajird-398	1	1	american	american	PROPN
ajird-398	1	2	journal	journal	PROPN
ajird-398	1	3	of	of	ADP
ajird-398	1	4	interdisciplinary	interdisciplinary	ADJ
ajird-398	1	5	research	research	NOUN
ajird-398	1	6	and	and	CCONJ
ajird-398	1	7	development	development	NOUN
ajird-398	1	8	issn	issn	PROPN
ajird-398	1	9	online	online	NOUN
ajird-398	1	10	:	:	PUNCT
ajird-398	1	11	2771	2771	NUM
ajird-398	1	12	-	-	SYM
ajird-398	1	13	8948	8948	NUM
ajird-398	1	14	website	website	NOUN
ajird-398	1	15	:	:	PUNCT
ajird-398	1	16	www.ajird.journalspark.org	www.ajird.journalspark.org	ADJ
ajird-398	1	17	volume	volume	NOUN
ajird-398	1	18	11	11	NUM
ajird-398	1	19	,	,	PUNCT
ajird-398	1	20	dec	dec	PROPN
ajird-398	1	21	.	.	PROPN
ajird-398	1	22	,	,	PUNCT
ajird-398	1	23	2022	2022	NUM
ajird-398	1	24	35	35	NUM
ajird-398	2	1	|	|	ADV
ajird-398	2	2	p	p	NOUN
ajird-398	2	3	a	a	DET
ajird-398	2	4	g	g	NOUN
ajird-398	2	5	e	e	X
ajird-398	2	6	asymptotic	asymptotic	ADJ
ajird-398	2	7	evaluation	evaluation	NOUN
ajird-398	2	8	of	of	ADP
ajird-398	2	9	parameterdependent	parameterdependent	NOUN
ajird-398	2	10	integrals	integral	NOUN
ajird-398	2	11	umirzakova	umirzakova	VERB
ajird-398	2	12	iroda	iroda	PROPN
ajird-398	2	13	3rd	3rd	ADJ
ajird-398	2	14	year	year	NOUN
ajird-398	2	15	student	student	NOUN
ajird-398	2	16	of	of	ADP
ajird-398	2	17	the	the	DET
ajird-398	2	18	faculty	faculty	NOUN
ajird-398	2	19	of	of	ADP
ajird-398	2	20	mathematics	mathematics	PROPN
ajird-398	2	21	of	of	ADP
ajird-398	2	22	samarkand	samarkand	PROPN
ajird-398	2	23	state	state	PROPN
ajird-398	2	24	university	university	PROPN
ajird-398	2	25	named	name	VERB
ajird-398	2	26	after	after	ADP
ajird-398	2	27	sharof	sharof	NOUN
ajird-398	3	1	rashidov	rashidov	PROPN
ajird-398	3	2	komilov	komilov	PROPN
ajird-398	3	3	abdulaziz	abdulaziz	PROPN
ajird-398	3	4	3rd	3rd	ADJ
ajird-398	3	5	year	year	NOUN
ajird-398	3	6	student	student	NOUN
ajird-398	3	7	of	of	ADP
ajird-398	3	8	the	the	DET
ajird-398	3	9	faculty	faculty	NOUN
ajird-398	3	10	of	of	ADP
ajird-398	3	11	mathematics	mathematics	PROPN
ajird-398	3	12	of	of	ADP
ajird-398	3	13	samarkand	samarkand	PROPN
ajird-398	3	14	state	state	PROPN
ajird-398	3	15	university	university	PROPN
ajird-398	3	16	named	name	VERB
ajird-398	3	17	after	after	ADP
ajird-398	3	18	sharof	sharof	NOUN
ajird-398	3	19	rashidov	rashidov	PROPN
ajird-398	3	20	abstract	abstract	ADV
ajird-398	3	21	this	this	DET
ajird-398	3	22	thesis	thesis	NOUN
ajird-398	3	23	presents	present	VERB
ajird-398	3	24	the	the	DET
ajird-398	3	25	methods	method	NOUN
ajird-398	3	26	of	of	ADP
ajird-398	3	27	calculating	calculate	VERB
ajird-398	3	28	integrals	integral	NOUN
ajird-398	3	29	that	that	PRON
ajird-398	3	30	are	be	AUX
ajird-398	3	31	approximate	approximate	ADJ
ajird-398	3	32	,	,	PUNCT
ajird-398	3	33	but	but	CCONJ
ajird-398	3	34	the	the	DET
ajird-398	3	35	calculation	calculation	NOUN
ajird-398	3	36	of	of	ADP
ajird-398	3	37	their	their	PRON
ajird-398	3	38	value	value	NOUN
ajird-398	3	39	is	be	AUX
ajird-398	3	40	quite	quite	ADV
ajird-398	3	41	complicated	complicated	ADJ
ajird-398	3	42	.	.	PUNCT
ajird-398	4	1	keywords	keyword	NOUN
ajird-398	4	2	:	:	PUNCT
ajird-398	4	3	parameter	parameter	NOUN
ajird-398	4	4	-	-	PUNCT
ajird-398	4	5	dependent	dependent	ADJ
ajird-398	4	6	integral	integral	ADJ
ajird-398	4	7	,	,	PUNCT
ajird-398	4	8	asymptotic	asymptotic	ADJ
ajird-398	4	9	estimation	estimation	NOUN
ajird-398	4	10	,	,	PUNCT
ajird-398	4	11	taylor	taylor	PROPN
ajird-398	4	12	series	series	PROPN
ajird-398	4	13	,	,	PUNCT
ajird-398	4	14	euler	euler	NOUN
ajird-398	4	15	integrals	integral	NOUN
ajird-398	4	16	.	.	PUNCT
ajird-398	5	1	when	when	SCONJ
ajird-398	5	2	constructing	construct	VERB
ajird-398	5	3	mathematical	mathematical	ADJ
ajird-398	5	4	models	model	NOUN
ajird-398	5	5	of	of	ADP
ajird-398	5	6	life	life	NOUN
ajird-398	5	7	problems	problem	NOUN
ajird-398	5	8	,	,	PUNCT
ajird-398	5	9	in	in	ADP
ajird-398	5	10	most	most	ADJ
ajird-398	5	11	cases	case	NOUN
ajird-398	5	12	,	,	PUNCT
ajird-398	5	13	it	it	PRON
ajird-398	5	14	is	be	AUX
ajird-398	5	15	important	important	ADJ
ajird-398	5	16	to	to	PART
ajird-398	5	17	know	know	VERB
ajird-398	5	18	the	the	DET
ajird-398	5	19	approximate	approximate	ADJ
ajird-398	5	20	value	value	NOUN
ajird-398	5	21	of	of	ADP
ajird-398	5	22	the	the	DET
ajird-398	5	23	solution	solution	NOUN
ajird-398	5	24	closest	close	ADJ
ajird-398	5	25	to	to	ADP
ajird-398	5	26	this	this	DET
ajird-398	5	27	solution	solution	NOUN
ajird-398	5	28	,	,	PUNCT
ajird-398	5	29	rather	rather	ADV
ajird-398	5	30	than	than	ADP
ajird-398	5	31	finding	find	VERB
ajird-398	5	32	the	the	DET
ajird-398	5	33	exact	exact	ADJ
ajird-398	5	34	value	value	NOUN
ajird-398	5	35	or	or	CCONJ
ajird-398	5	36	values	value	NOUN
ajird-398	5	37	of	of	ADP
ajird-398	5	38	the	the	DET
ajird-398	5	39	problem	problem	NOUN
ajird-398	5	40	solution	solution	NOUN
ajird-398	5	41	.	.	PUNCT
ajird-398	6	1	this	this	DET
ajird-398	6	2	thesis	thesis	NOUN
ajird-398	6	3	considers	consider	VERB
ajird-398	6	4	the	the	DET
ajird-398	6	5	issue	issue	NOUN
ajird-398	6	6	of	of	ADP
ajird-398	6	7	approximate	approximate	ADJ
ajird-398	6	8	calculation	calculation	NOUN
ajird-398	6	9	of	of	ADP
ajird-398	6	10	integrals	integral	NOUN
ajird-398	6	11	depending	depend	VERB
ajird-398	6	12	on	on	ADP
ajird-398	6	13	the	the	DET
ajird-398	6	14	parameter	parameter	NOUN
ajird-398	6	15	or	or	CCONJ
ajird-398	6	16	asymptotic	asymptotic	ADJ
ajird-398	6	17	estimation	estimation	NOUN
ajird-398	6	18	of	of	ADP
ajird-398	6	19	the	the	DET
ajird-398	6	20	parameter	parameter	NOUN
ajird-398	6	21	,	,	PUNCT
ajird-398	6	22	which	which	PRON
ajird-398	6	23	plays	play	VERB
ajird-398	6	24	an	an	DET
ajird-398	6	25	important	important	ADJ
ajird-398	6	26	role	role	NOUN
ajird-398	6	27	in	in	ADP
ajird-398	6	28	solving	solve	VERB
ajird-398	6	29	this	this	DET
ajird-398	6	30	type	type	NOUN
ajird-398	6	31	of	of	ADP
ajird-398	6	32	problems	problem	NOUN
ajird-398	6	33	.	.	PUNCT
ajird-398	7	1	in	in	ADP
ajird-398	7	2	mathematics	mathematic	NOUN
ajird-398	7	3	,	,	PUNCT
ajird-398	7	4	an	an	DET
ajird-398	7	5	analytic	analytic	ADJ
ajird-398	7	6	function	function	NOUN
ajird-398	7	7	is	be	AUX
ajird-398	7	8	a	a	DET
ajird-398	7	9	function	function	NOUN
ajird-398	7	10	that	that	PRON
ajird-398	7	11	is	be	AUX
ajird-398	7	12	locally	locally	ADV
ajird-398	7	13	given	give	VERB
ajird-398	7	14	by	by	ADP
ajird-398	7	15	a	a	DET
ajird-398	7	16	convergent	convergent	NOUN
ajird-398	7	17	power	power	NOUN
ajird-398	7	18	series	series	NOUN
ajird-398	7	19	.	.	PUNCT
ajird-398	8	1	first	first	ADV
ajird-398	8	2	,	,	PUNCT
ajird-398	8	3	we	we	PRON
ajird-398	8	4	present	present	VERB
ajird-398	8	5	the	the	DET
ajird-398	8	6	following	follow	VERB
ajird-398	8	7	theorem	theorem	ADJ
ajird-398	8	8	:	:	PUNCT
ajird-398	8	9	theorem	theorem	ADJ
ajird-398	8	10	.	.	PUNCT
ajird-398	9	1	if	if	SCONJ
ajird-398	9	2	every	every	DET
ajird-398	9	3	term	term	NOUN
ajird-398	9	4	of	of	ADP
ajird-398	9	5	the	the	DET
ajird-398	9	6	series	series	NOUN
ajird-398	9	7	∑	∑	PROPN
ajird-398	9	8	𝑢𝑛(𝑥)∞	𝑢𝑛(𝑥)∞	NOUN
ajird-398	9	9	𝑛=1	𝑛=1	NOUN
ajird-398	9	10	is	be	AUX
ajird-398	9	11	continuous	continuous	ADJ
ajird-398	9	12	in	in	ADP
ajird-398	9	13	the	the	DET
ajird-398	9	14	segment	segment	NOUN
ajird-398	9	15	[	[	X
ajird-398	9	16	𝑎	𝑎	X
ajird-398	9	17	,	,	PUNCT
ajird-398	9	18	𝑏	𝑏	NOUN
ajird-398	9	19	]	]	X
ajird-398	9	20	,	,	PUNCT
ajird-398	9	21	and	and	CCONJ
ajird-398	9	22	this	this	DET
ajird-398	9	23	series	series	NOUN
ajird-398	9	24	is	be	AUX
ajird-398	9	25	uniformly	uniformly	ADV
ajird-398	9	26	convergent	convergent	NOUN
ajird-398	9	27	in	in	ADP
ajird-398	9	28	this	this	DET
ajird-398	9	29	segment	segment	NOUN
ajird-398	9	30	and	and	CCONJ
ajird-398	9	31	𝑆(𝑥	𝑆(𝑥	NUM
ajird-398	9	32	)	)	PUNCT
ajird-398	9	33	is	be	AUX
ajird-398	9	34	a	a	DET
ajird-398	9	35	sum	sum	NOUN
ajird-398	9	36	of	of	ADP
ajird-398	9	37	series	series	NOUN
ajird-398	9	38	,	,	PUNCT
ajird-398	9	39	then	then	ADV
ajird-398	9	40	the	the	DET
ajird-398	9	41	following	follow	VERB
ajird-398	9	42	equality	equality	NOUN
ajird-398	9	43	holds	hold	VERB
ajird-398	9	44	:	:	PUNCT
ajird-398	9	45	∫	∫	PROPN
ajird-398	9	46	𝑆(𝑥	𝑆(𝑥	NUM
ajird-398	9	47	)	)	PUNCT
ajird-398	9	48	𝑏	𝑏	NOUN
ajird-398	9	49	𝑎	𝑎	NOUN
ajird-398	9	50	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	9	51	=	=	PUNCT
ajird-398	9	52	∑	∑	PROPN
ajird-398	9	53	∫	∫	PROPN
ajird-398	9	54	𝑢𝑛(𝑥	𝑢𝑛(𝑥	NUM
ajird-398	9	55	)	)	PUNCT
ajird-398	9	56	𝑏	𝑏	DET
ajird-398	9	57	𝑎	𝑎	PROPN
ajird-398	9	58	𝑑𝑥∞	𝑑𝑥∞	PROPN
ajird-398	9	59	𝑛=1	𝑛=1	PROPN
ajird-398	9	60	.	.	PUNCT
ajird-398	10	1	let	let	VERB
ajird-398	10	2	the	the	DET
ajird-398	10	3	following	follow	VERB
ajird-398	10	4	integral	integral	ADJ
ajird-398	10	5	depending	depend	VERB
ajird-398	10	6	on	on	ADP
ajird-398	10	7	parameters	parameter	NOUN
ajird-398	10	8	𝑝	𝑝	PROPN
ajird-398	10	9	and	and	CCONJ
ajird-398	10	10	𝑞	𝑞	X
ajird-398	10	11	be	be	AUX
ajird-398	10	12	given	give	VERB
ajird-398	10	13	:	:	PUNCT
ajird-398	10	14	φ(𝑝	φ(𝑝	ADJ
ajird-398	10	15	,	,	PUNCT
ajird-398	10	16	𝑞	𝑞	X
ajird-398	10	17	)	)	PUNCT
ajird-398	10	18	=	=	SYM
ajird-398	10	19	∫	∫	PROPN
ajird-398	10	20	𝑓(𝑥	𝑓(𝑥	NOUN
ajird-398	10	21	,	,	PUNCT
ajird-398	10	22	𝑝	𝑝	NOUN
ajird-398	10	23	)	)	PUNCT
ajird-398	10	24	∙	∙	PROPN
ajird-398	10	25	𝑔(𝑥	𝑔(𝑥	PROPN
ajird-398	10	26	,	,	PUNCT
ajird-398	10	27	𝑞	𝑞	NOUN
ajird-398	10	28	)	)	PUNCT
ajird-398	10	29	𝑏	𝑏	PROPN
ajird-398	10	30	𝑎	𝑎	NOUN
ajird-398	10	31	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	10	32	(	(	PUNCT
ajird-398	10	33	1	1	NUM
ajird-398	10	34	)	)	PUNCT
ajird-398	10	35	let	let	VERB
ajird-398	10	36	us	we	PRON
ajird-398	10	37	assume	assume	VERB
ajird-398	10	38	that	that	SCONJ
ajird-398	10	39	the	the	DET
ajird-398	10	40	functions	function	NOUN
ajird-398	10	41	𝑓(𝑥	𝑓(𝑥	NOUN
ajird-398	10	42	,	,	PUNCT
ajird-398	10	43	𝑝	𝑝	NOUN
ajird-398	10	44	)	)	PUNCT
ajird-398	10	45	,	,	PUNCT
ajird-398	10	46	𝑔(𝑥	𝑔(𝑥	PROPN
ajird-398	10	47	,	,	PUNCT
ajird-398	10	48	𝑞	𝑞	NOUN
ajird-398	10	49	)	)	PUNCT
ajird-398	10	50	are	be	AUX
ajird-398	10	51	analytic	analytic	ADJ
ajird-398	10	52	in	in	ADP
ajird-398	10	53	some	some	DET
ajird-398	10	54	interval	interval	NOUN
ajird-398	10	55	(	(	PUNCT
ajird-398	10	56	−𝜌	−𝜌	PROPN
ajird-398	10	57	,	,	PUNCT
ajird-398	10	58	𝜌	𝜌	ADP
ajird-398	10	59	)	)	PUNCT
ajird-398	10	60	,	,	PUNCT
ajird-398	10	61	and	and	CCONJ
ajird-398	10	62	this	this	DET
ajird-398	10	63	condition	condition	NOUN
ajird-398	10	64	(	(	PUNCT
ajird-398	10	65	𝑎	𝑎	NOUN
ajird-398	10	66	,	,	PUNCT
ajird-398	10	67	𝑏	𝑏	NOUN
ajird-398	10	68	)	)	PUNCT
ajird-398	10	69	⊂	⊂	PROPN
ajird-398	10	70	(	(	PUNCT
ajird-398	10	71	−𝜌	−𝜌	PROPN
ajird-398	10	72	,	,	PUNCT
ajird-398	10	73	𝜌	𝜌	X
ajird-398	10	74	)	)	PUNCT
ajird-398	10	75	is	be	AUX
ajird-398	10	76	hold	hold	NOUN
ajird-398	10	77	.	.	PUNCT
ajird-398	11	1	then	then	ADV
ajird-398	11	2	the	the	DET
ajird-398	11	3	following	follow	VERB
ajird-398	11	4	equations	equation	NOUN
ajird-398	11	5	hold	hold	VERB
ajird-398	11	6	for	for	ADP
ajird-398	11	7	∀𝑥𝜖(𝑎	∀𝑥𝜖(𝑎	PROPN
ajird-398	11	8	,	,	PUNCT
ajird-398	11	9	𝑏	𝑏	NOUN
ajird-398	11	10	):	):	PUNCT
ajird-398	11	11	https://en.wikipedia.org/wiki/mathematics	https://en.wikipedia.org/wiki/mathematics	PROPN
ajird-398	11	12	https://en.wikipedia.org/wiki/function_(mathematics	https://en.wikipedia.org/wiki/function_(mathematic	NOUN
ajird-398	11	13	)	)	PUNCT
ajird-398	11	14	https://en.wikipedia.org/wiki/convergent_series	https://en.wikipedia.org/wiki/convergent_serie	NOUN
ajird-398	11	15	https://en.wikipedia.org/wiki/power_series	https://en.wikipedia.org/wiki/power_series	PROPN
ajird-398	11	16	https://en.wikipedia.org/wiki/power_series	https://en.wikipedia.org/wiki/power_series	PROPN
ajird-398	11	17	american	american	PROPN
ajird-398	11	18	journal	journal	PROPN
ajird-398	11	19	of	of	ADP
ajird-398	11	20	interdisciplinary	interdisciplinary	ADJ
ajird-398	11	21	research	research	NOUN
ajird-398	11	22	and	and	CCONJ
ajird-398	11	23	development	development	NOUN
ajird-398	11	24	issn	issn	PROPN
ajird-398	11	25	online	online	NOUN
ajird-398	11	26	:	:	PUNCT
ajird-398	11	27	2771	2771	NUM
ajird-398	11	28	-	-	SYM
ajird-398	11	29	8948	8948	NUM
ajird-398	11	30	website	website	NOUN
ajird-398	11	31	:	:	PUNCT
ajird-398	11	32	www.ajird.journalspark.org	www.ajird.journalspark.org	ADJ
ajird-398	11	33	volume	volume	NOUN
ajird-398	11	34	11	11	NUM
ajird-398	11	35	,	,	PUNCT
ajird-398	11	36	dec	dec	PROPN
ajird-398	11	37	.	.	PROPN
ajird-398	11	38	,	,	PUNCT
ajird-398	11	39	2022	2022	NUM
ajird-398	11	40	36	36	NUM
ajird-398	12	1	|	|	ADV
ajird-398	12	2	p	p	ADP
ajird-398	12	3	a	a	DET
ajird-398	12	4	g	g	NOUN
ajird-398	12	5	e	e	NOUN
ajird-398	12	6	𝑓(𝑥	𝑓(𝑥	NOUN
ajird-398	12	7	,	,	PUNCT
ajird-398	12	8	𝑝	𝑝	NOUN
ajird-398	12	9	)	)	PUNCT
ajird-398	12	10	=	=	PUNCT
ajird-398	13	1	∑	∑	PUNCT
ajird-398	13	2	𝑓𝑛(0	𝑓𝑛(0	ADJ
ajird-398	13	3	,	,	PUNCT
ajird-398	13	4	𝑝	𝑝	NOUN
ajird-398	13	5	)	)	PUNCT
ajird-398	13	6	𝑛	𝑛	NOUN
ajird-398	13	7	!	!	PUNCT
ajird-398	14	1	∙	∙	NOUN
ajird-398	14	2	𝑥𝑛	𝑥𝑛	PROPN
ajird-398	14	3	∞	∞	NUM
ajird-398	14	4	𝑛=0	𝑛=0	PROPN
ajird-398	14	5	𝑔(𝑥	𝑔(𝑥	PROPN
ajird-398	14	6	,	,	PUNCT
ajird-398	14	7	𝑞	𝑞	X
ajird-398	14	8	)	)	PUNCT
ajird-398	14	9	=	=	SYM
ajird-398	14	10	∑	∑	PUNCT
ajird-398	14	11	𝑓𝑛(0	𝑓𝑛(0	NOUN
ajird-398	14	12	,	,	PUNCT
ajird-398	14	13	𝑞	𝑞	NOUN
ajird-398	14	14	)	)	PUNCT
ajird-398	14	15	𝑛	𝑛	VERB
ajird-398	14	16	!	!	PUNCT
ajird-398	15	1	∙	∙	NOUN
ajird-398	15	2	𝑥𝑛	𝑥𝑛	PROPN
ajird-398	15	3	∞	∞	NUM
ajird-398	15	4	𝑛=0	𝑛=0	NOUN
ajird-398	15	5	using	use	VERB
ajird-398	15	6	these	these	DET
ajird-398	15	7	equations	equation	NOUN
ajird-398	15	8	and	and	CCONJ
ajird-398	15	9	the	the	DET
ajird-398	15	10	above	above	ADJ
ajird-398	15	11	theorem	theorem	NOUN
ajird-398	15	12	,	,	PUNCT
ajird-398	15	13	we	we	PRON
ajird-398	15	14	can	can	AUX
ajird-398	15	15	write	write	VERB
ajird-398	15	16	integral	integral	ADJ
ajird-398	15	17	(	(	PUNCT
ajird-398	15	18	1	1	NUM
ajird-398	15	19	)	)	PUNCT
ajird-398	15	20	in	in	ADP
ajird-398	15	21	the	the	DET
ajird-398	15	22	following	follow	VERB
ajird-398	15	23	form	form	NOUN
ajird-398	15	24	:	:	PUNCT
ajird-398	15	25	φ(𝑝	φ(𝑝	X
ajird-398	15	26	,	,	PUNCT
ajird-398	15	27	𝑞	𝑞	X
ajird-398	15	28	)	)	PUNCT
ajird-398	15	29	=	=	SYM
ajird-398	15	30	∫	∫	PROPN
ajird-398	15	31	∑	∑	PROPN
ajird-398	15	32	𝑓𝑛(0	𝑓𝑛(0	NOUN
ajird-398	15	33	,	,	PUNCT
ajird-398	15	34	𝑝	𝑝	NOUN
ajird-398	15	35	)	)	PUNCT
ajird-398	15	36	𝑛	𝑛	NOUN
ajird-398	15	37	!	!	PUNCT
ajird-398	16	1	∙	∙	NOUN
ajird-398	16	2	𝑥𝑛	𝑥𝑛	PROPN
ajird-398	17	1	∙	∙	PROPN
ajird-398	17	2	𝑔(𝑥	𝑔(𝑥	PROPN
ajird-398	17	3	,	,	PUNCT
ajird-398	17	4	𝑞	𝑞	NOUN
ajird-398	17	5	)	)	PUNCT
ajird-398	17	6	∞	∞	NUM
ajird-398	18	1	𝑛=0	𝑛=0	PROPN
ajird-398	19	1	𝑏	𝑏	PRON
ajird-398	19	2	𝑎	𝑎	NOUN
ajird-398	19	3	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	19	4	=	=	SYM
ajird-398	19	5	∑	∑	PUNCT
ajird-398	19	6	𝑓𝑛(0	𝑓𝑛(0	ADJ
ajird-398	19	7	,	,	PUNCT
ajird-398	19	8	𝑝	𝑝	NOUN
ajird-398	19	9	)	)	PUNCT
ajird-398	19	10	𝑛	𝑛	PROPN
ajird-398	19	11	!	!	PROPN
ajird-398	19	12	∫𝑔(𝑥	∫𝑔(𝑥	PROPN
ajird-398	19	13	,	,	PUNCT
ajird-398	19	14	𝑞	𝑞	X
ajird-398	19	15	)	)	PUNCT
ajird-398	19	16	∙	∙	PROPN
ajird-398	19	17	𝑥𝑛	𝑥𝑛	VERB
ajird-398	19	18	𝑏	𝑏	ADV
ajird-398	19	19	𝑎	𝑎	NOUN
ajird-398	19	20	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	19	21	∞	∞	NUM
ajird-398	19	22	𝑛=0	𝑛=0	PROPN
ajird-398	19	23	,	,	PUNCT
ajird-398	19	24	φ(𝑝	φ(𝑝	PROPN
ajird-398	19	25	,	,	PUNCT
ajird-398	19	26	𝑞	𝑞	X
ajird-398	19	27	)	)	PUNCT
ajird-398	19	28	=	=	SYM
ajird-398	19	29	∫	∫	PROPN
ajird-398	19	30	∑	∑	PROPN
ajird-398	19	31	𝑔𝑛(0	𝑔𝑛(0	PROPN
ajird-398	19	32	,	,	PUNCT
ajird-398	19	33	𝑞	𝑞	NOUN
ajird-398	19	34	)	)	PUNCT
ajird-398	19	35	𝑛	𝑛	VERB
ajird-398	19	36	!	!	PUNCT
ajird-398	20	1	∙	∙	NOUN
ajird-398	20	2	𝑥𝑛	𝑥𝑛	VERB
ajird-398	21	1	∙	∙	PROPN
ajird-398	21	2	𝑓(𝑥	𝑓(𝑥	NOUN
ajird-398	21	3	,	,	PUNCT
ajird-398	21	4	𝑝	𝑝	NOUN
ajird-398	21	5	)	)	PUNCT
ajird-398	21	6	∞	∞	NUM
ajird-398	22	1	𝑛=0	𝑛=0	PROPN
ajird-398	23	1	𝑏	𝑏	PRON
ajird-398	23	2	𝑎	𝑎	NOUN
ajird-398	23	3	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	23	4	=	=	PUNCT
ajird-398	23	5	∑	∑	PUNCT
ajird-398	23	6	𝑔𝑛(0	𝑔𝑛(0	PROPN
ajird-398	23	7	,	,	PUNCT
ajird-398	23	8	𝑞	𝑞	NOUN
ajird-398	23	9	)	)	PUNCT
ajird-398	23	10	𝑛	𝑛	PROPN
ajird-398	23	11	!	!	PROPN
ajird-398	23	12	∫	∫	PROPN
ajird-398	23	13	𝑓(𝑥	𝑓(𝑥	NOUN
ajird-398	23	14	,	,	PUNCT
ajird-398	23	15	𝑝	𝑝	NOUN
ajird-398	23	16	)	)	PUNCT
ajird-398	23	17	∙	∙	PROPN
ajird-398	23	18	𝑥𝑛	𝑥𝑛	VERB
ajird-398	23	19	𝑏	𝑏	ADV
ajird-398	23	20	𝑎	𝑎	NOUN
ajird-398	23	21	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	23	22	∞	∞	NUM
ajird-398	23	23	𝑛=0	𝑛=0	PROPN
ajird-398	23	24	.	.	PUNCT
ajird-398	24	1	here	here	ADV
ajird-398	24	2	𝑓𝑛(0,𝑝	𝑓𝑛(0,𝑝	VERB
ajird-398	24	3	)	)	PUNCT
ajird-398	24	4	𝑛	𝑛	PROPN
ajird-398	24	5	!	!	NOUN
ajird-398	24	6	and	and	CCONJ
ajird-398	24	7	𝑔𝑛(0,𝑞	𝑔𝑛(0,𝑞	NOUN
ajird-398	24	8	)	)	PUNCT
ajird-398	24	9	𝑛	𝑛	NOUN
ajird-398	24	10	!	!	NOUN
ajird-398	24	11	are	be	AUX
ajird-398	24	12	the	the	DET
ajird-398	24	13	taylor	taylor	PROPN
ajird-398	24	14	coefficients	coefficient	NOUN
ajird-398	24	15	of	of	ADP
ajird-398	24	16	the	the	DET
ajird-398	24	17	functions	function	NOUN
ajird-398	24	18	𝑓(𝑥	𝑓(𝑥	NOUN
ajird-398	24	19	,	,	PUNCT
ajird-398	24	20	𝑝	𝑝	NOUN
ajird-398	24	21	)	)	PUNCT
ajird-398	24	22	and	and	CCONJ
ajird-398	24	23	𝑔(𝑥	𝑔(𝑥	PROPN
ajird-398	24	24	,	,	PUNCT
ajird-398	24	25	𝑞	𝑞	NOUN
ajird-398	24	26	)	)	PUNCT
ajird-398	24	27	,	,	PUNCT
ajird-398	24	28	respectively	respectively	ADV
ajird-398	24	29	.	.	PUNCT
ajird-398	25	1	for	for	ADP
ajird-398	25	2	convenience	convenience	NOUN
ajird-398	25	3	,	,	PUNCT
ajird-398	25	4	we	we	PRON
ajird-398	25	5	choose	choose	VERB
ajird-398	25	6	one	one	NUM
ajird-398	25	7	of	of	ADP
ajird-398	25	8	the	the	DET
ajird-398	25	9	integrals	integral	NOUN
ajird-398	25	10	∫	∫	PROPN
ajird-398	26	1	𝑥𝑛𝑓(𝑥	𝑥𝑛𝑓(𝑥	PROPN
ajird-398	26	2	,	,	PUNCT
ajird-398	26	3	𝑝	𝑝	NOUN
ajird-398	26	4	)	)	PUNCT
ajird-398	27	1	𝑏	𝑏	NOUN
ajird-398	27	2	𝑎	𝑎	NOUN
ajird-398	27	3	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	27	4	and	and	CCONJ
ajird-398	27	5	∫	∫	PROPN
ajird-398	27	6	𝑥𝑛𝑔(𝑥	𝑥𝑛𝑔(𝑥	PROPN
ajird-398	27	7	,	,	PUNCT
ajird-398	27	8	𝑞	𝑞	PROPN
ajird-398	27	9	)	)	PUNCT
ajird-398	27	10	𝑏	𝑏	PROPN
ajird-398	27	11	𝑎	𝑎	NOUN
ajird-398	27	12	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	27	13	in	in	ADP
ajird-398	27	14	such	such	DET
ajird-398	27	15	a	a	DET
ajird-398	27	16	way	way	NOUN
ajird-398	27	17	that	that	PRON
ajird-398	27	18	to	to	PART
ajird-398	27	19	calculate	calculate	VERB
ajird-398	27	20	the	the	DET
ajird-398	27	21	chosen	choose	VERB
ajird-398	27	22	integral	integral	ADJ
ajird-398	27	23	be	be	AUX
ajird-398	27	24	easy	easy	ADJ
ajird-398	27	25	.	.	PUNCT
ajird-398	28	1	suppose	suppose	VERB
ajird-398	28	2	that	that	SCONJ
ajird-398	28	3	the	the	DET
ajird-398	28	4	equality∫	equality∫	NOUN
ajird-398	28	5	𝑥𝑛𝑓(𝑥	𝑥𝑛𝑓(𝑥	PROPN
ajird-398	28	6	,	,	PUNCT
ajird-398	28	7	𝑝	𝑝	NOUN
ajird-398	28	8	)	)	PUNCT
ajird-398	28	9	𝑏	𝑏	NOUN
ajird-398	28	10	𝑎	𝑎	NOUN
ajird-398	28	11	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	28	12	=	=	SYM
ajird-398	28	13	𝜓(𝑛	𝜓(𝑛	PROPN
ajird-398	28	14	,	,	PUNCT
ajird-398	28	15	𝑝	𝑝	NOUN
ajird-398	28	16	)	)	PUNCT
ajird-398	28	17	is	be	AUX
ajird-398	28	18	hold	hold	NOUN
ajird-398	28	19	,	,	PUNCT
ajird-398	28	20	then	then	ADV
ajird-398	28	21	∫𝑓(𝑥	∫𝑓(𝑥	NOUN
ajird-398	28	22	,	,	PUNCT
ajird-398	28	23	𝑝	𝑝	NOUN
ajird-398	28	24	)	)	PUNCT
ajird-398	28	25	∙	∙	PROPN
ajird-398	28	26	𝑔(𝑥	𝑔(𝑥	PROPN
ajird-398	28	27	,	,	PUNCT
ajird-398	28	28	𝑞	𝑞	NOUN
ajird-398	28	29	)	)	PUNCT
ajird-398	28	30	𝑏	𝑏	PROPN
ajird-398	28	31	𝑎	𝑎	NOUN
ajird-398	28	32	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	28	33	=	=	PUNCT
ajird-398	28	34	∫	∫	PROPN
ajird-398	28	35	∑	∑	PROPN
ajird-398	28	36	𝑔𝑛(0	𝑔𝑛(0	PROPN
ajird-398	28	37	,	,	PUNCT
ajird-398	28	38	𝑞	𝑞	NOUN
ajird-398	28	39	)	)	PUNCT
ajird-398	28	40	𝑛	𝑛	VERB
ajird-398	28	41	!	!	PUNCT
ajird-398	29	1	∙	∙	PROPN
ajird-398	29	2	𝑥𝑛𝑓(𝑥	𝑥𝑛𝑓(𝑥	PROPN
ajird-398	29	3	,	,	PUNCT
ajird-398	29	4	𝑝	𝑝	NOUN
ajird-398	29	5	)	)	PUNCT
ajird-398	29	6	∞	∞	NUM
ajird-398	30	1	𝑛=0	𝑛=0	PROPN
ajird-398	31	1	𝑏	𝑏	PRON
ajird-398	31	2	𝑎	𝑎	NOUN
ajird-398	31	3	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	31	4	=	=	SYM
ajird-398	31	5	=	=	SYM
ajird-398	31	6	∑	∑	PUNCT
ajird-398	31	7	𝑔𝑛(0	𝑔𝑛(0	PROPN
ajird-398	31	8	,	,	PUNCT
ajird-398	31	9	𝑞	𝑞	NOUN
ajird-398	31	10	)	)	PUNCT
ajird-398	31	11	𝑛	𝑛	PROPN
ajird-398	31	12	!	!	NOUN
ajird-398	31	13	∫𝑓(𝑥	∫𝑓(𝑥	NOUN
ajird-398	31	14	,	,	PUNCT
ajird-398	31	15	𝑝	𝑝	NOUN
ajird-398	31	16	)	)	PUNCT
ajird-398	32	1	∙	∙	PROPN
ajird-398	32	2	𝑥𝑛	𝑥𝑛	VERB
ajird-398	33	1	𝑏	𝑏	ADV
ajird-398	33	2	𝑎	𝑎	NOUN
ajird-398	33	3	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	33	4	∞	∞	NUM
ajird-398	33	5	𝑛=0	𝑛=0	PROPN
ajird-398	33	6	=	=	SYM
ajird-398	33	7	∑	∑	PUNCT
ajird-398	33	8	𝑔𝑛(0	𝑔𝑛(0	PROPN
ajird-398	33	9	,	,	PUNCT
ajird-398	33	10	𝑞	𝑞	NOUN
ajird-398	33	11	)	)	PUNCT
ajird-398	33	12	𝑛	𝑛	VERB
ajird-398	33	13	!	!	PUNCT
ajird-398	34	1	∙	∙	PROPN
ajird-398	34	2	𝜓(𝑛	𝜓(𝑛	PROPN
ajird-398	34	3	,	,	PUNCT
ajird-398	34	4	𝑝	𝑝	NOUN
ajird-398	34	5	)	)	PUNCT
ajird-398	34	6	∞	∞	NUM
ajird-398	34	7	𝑛=0	𝑛=0	PROPN
ajird-398	34	8	.	.	PUNCT
ajird-398	35	1	if	if	SCONJ
ajird-398	35	2	we	we	PRON
ajird-398	35	3	choose	choose	VERB
ajird-398	35	4	𝑛	𝑛	X
ajird-398	35	5	<	<	X
ajird-398	35	6	𝑛0	𝑛0	VERB
ajird-398	35	7	in	in	ADP
ajird-398	35	8	this	this	DET
ajird-398	35	9	equation	equation	NOUN
ajird-398	35	10	,	,	PUNCT
ajird-398	35	11	we	we	PRON
ajird-398	35	12	arrive	arrive	VERB
ajird-398	35	13	at	at	ADP
ajird-398	35	14	the	the	DET
ajird-398	35	15	following	following	ADJ
ajird-398	35	16	approximate	approximate	ADJ
ajird-398	35	17	equation	equation	NOUN
ajird-398	35	18	:	:	PUNCT
ajird-398	35	19	∫	∫	PROPN
ajird-398	35	20	𝑓(𝑥	𝑓(𝑥	NOUN
ajird-398	35	21	,	,	PUNCT
ajird-398	35	22	𝑝	𝑝	NOUN
ajird-398	35	23	)	)	PUNCT
ajird-398	35	24	∙	∙	PROPN
ajird-398	35	25	𝑔(𝑥	𝑔(𝑥	PROPN
ajird-398	35	26	,	,	PUNCT
ajird-398	35	27	𝑞	𝑞	NOUN
ajird-398	35	28	)	)	PUNCT
ajird-398	35	29	𝑏	𝑏	PROPN
ajird-398	35	30	𝑎	𝑎	NOUN
ajird-398	35	31	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	35	32	≈	≈	PROPN
ajird-398	35	33	∑	∑	PROPN
ajird-398	35	34	𝑔𝑛(0	𝑔𝑛(0	PROPN
ajird-398	35	35	,	,	PUNCT
ajird-398	35	36	𝑞	𝑞	NOUN
ajird-398	35	37	)	)	PUNCT
ajird-398	35	38	𝑛	𝑛	VERB
ajird-398	35	39	!	!	PUNCT
ajird-398	36	1	∙	∙	PROPN
ajird-398	36	2	𝜓(𝑛	𝜓(𝑛	PROPN
ajird-398	36	3	,	,	PUNCT
ajird-398	36	4	𝑝	𝑝	NOUN
ajird-398	36	5	)	)	PUNCT
ajird-398	36	6	𝑛0	𝑛0	VERB
ajird-398	36	7	𝑛=0	𝑛=0	X
ajird-398	36	8	it	it	PRON
ajird-398	36	9	is	be	AUX
ajird-398	36	10	known	know	VERB
ajird-398	36	11	that	that	SCONJ
ajird-398	36	12	the	the	DET
ajird-398	36	13	larger	large	ADJ
ajird-398	36	14	the	the	DET
ajird-398	36	15	number	number	NOUN
ajird-398	36	16	𝑛0	𝑛0	VERB
ajird-398	36	17	can	can	AUX
ajird-398	36	18	be	be	AUX
ajird-398	36	19	,	,	PUNCT
ajird-398	36	20	the	the	PRON
ajird-398	36	21	closer	close	ADV
ajird-398	36	22	the	the	DET
ajird-398	36	23	found	find	VERB
ajird-398	36	24	value	value	NOUN
ajird-398	36	25	is	be	AUX
ajird-398	36	26	to	to	ADP
ajird-398	36	27	the	the	DET
ajird-398	36	28	value	value	NOUN
ajird-398	36	29	of	of	ADP
ajird-398	36	30	the	the	DET
ajird-398	36	31	given	give	VERB
ajird-398	36	32	integral	integral	NOUN
ajird-398	36	33	.	.	PUNCT
ajird-398	37	1	some	some	DET
ajird-398	37	2	methods	method	NOUN
ajird-398	37	3	of	of	ADP
ajird-398	37	4	calculating	calculate	VERB
ajird-398	37	5	integrals	integral	NOUN
ajird-398	37	6	depending	depend	VERB
ajird-398	37	7	on	on	ADP
ajird-398	37	8	parameter	parameter	NOUN
ajird-398	37	9	are	be	AUX
ajird-398	37	10	given	give	VERB
ajird-398	37	11	below	below	ADP
ajird-398	37	12	:	:	PUNCT
ajird-398	37	13	problem	problem	NOUN
ajird-398	37	14	1	1	NUM
ajird-398	37	15	.	.	PUNCT
ajird-398	37	16	evaluate	evaluate	VERB
ajird-398	37	17	the	the	DET
ajird-398	37	18	value	value	NOUN
ajird-398	37	19	of	of	ADP
ajird-398	37	20	the	the	DET
ajird-398	37	21	following	follow	VERB
ajird-398	37	22	integral	integral	ADJ
ajird-398	37	23	:	:	PUNCT
ajird-398	37	24	𝐼(𝑟	𝐼(𝑟	NUM
ajird-398	37	25	)	)	PUNCT
ajird-398	37	26	=	=	SYM
ajird-398	38	1	∫	∫	PROPN
ajird-398	39	1	𝑒𝑚𝑟∙cos𝑥𝜋	𝑒𝑚𝑟∙cos𝑥𝜋	CCONJ
ajird-398	39	2	0	0	NUM
ajird-398	39	3	𝑑𝑥.	𝑑𝑥.	NOUN
ajird-398	39	4	solution	solution	NOUN
ajird-398	39	5	.	.	PUNCT
ajird-398	40	1	first	first	ADV
ajird-398	40	2	,	,	PUNCT
ajird-398	40	3	we	we	PRON
ajird-398	40	4	split	split	VERB
ajird-398	40	5	this	this	DET
ajird-398	40	6	integral	integral	ADJ
ajird-398	40	7	into	into	ADP
ajird-398	40	8	two	two	NUM
ajird-398	40	9	integrals	integral	NOUN
ajird-398	40	10	,	,	PUNCT
ajird-398	40	11	and	and	CCONJ
ajird-398	40	12	then	then	ADV
ajird-398	40	13	get	get	VERB
ajird-398	40	14	the	the	DET
ajird-398	40	15	substitution	substitution	NOUN
ajird-398	40	16	:	:	PUNCT
ajird-398	40	17	𝐼(𝑟	𝐼(𝑟	NUM
ajird-398	40	18	)	)	PUNCT
ajird-398	40	19	=	=	SYM
ajird-398	41	1	∫	∫	PROPN
ajird-398	42	1	𝑒𝑚𝑟∙cos𝑥	𝑒𝑚𝑟∙cos𝑥	INTJ
ajird-398	42	2	𝜋	𝜋	NOUN
ajird-398	42	3	0	0	PUNCT
ajird-398	42	4	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	42	5	=	=	SYM
ajird-398	42	6	∫	∫	NOUN
ajird-398	43	1	𝑒𝑚𝑟∙cos𝑥	𝑒𝑚𝑟∙cos𝑥	NOUN
ajird-398	43	2	𝜋	𝜋	NOUN
ajird-398	43	3	2	2	NUM
ajird-398	43	4	0	0	NUM
ajird-398	43	5	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	43	6	+	+	NOUN
ajird-398	43	7	∫	∫	PROPN
ajird-398	44	1	𝑒𝑚𝑟∙cos𝑥	𝑒𝑚𝑟∙cos𝑥	NOUN
ajird-398	44	2	𝜋	𝜋	NOUN
ajird-398	44	3	𝜋	𝜋	NOUN
ajird-398	44	4	2	2	NUM
ajird-398	44	5	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	44	6	=	=	SYM
ajird-398	44	7	[	[	PUNCT
ajird-398	44	8	𝑥	𝑥	X
ajird-398	44	9	=	=	SYM
ajird-398	44	10	𝑡	𝑡	PROPN
ajird-398	44	11	+	+	X
ajird-398	44	12	𝜋	𝜋	NOUN
ajird-398	44	13	2	2	NUM
ajird-398	44	14	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	44	15	=	=	PUNCT
ajird-398	44	16	𝑑𝑡	𝑑𝑡	ADP
ajird-398	44	17	𝑥	𝑥	NOUN
ajird-398	44	18	=	=	PUNCT
ajird-398	44	19	𝜋	𝜋	NOUN
ajird-398	44	20	2	2	NUM
ajird-398	44	21	→	→	SYM
ajird-398	44	22	𝑡	𝑡	NOUN
ajird-398	44	23	=	=	SYM
ajird-398	44	24	0	0	PUNCT
ajird-398	45	1	𝑥	𝑥	NOUN
ajird-398	45	2	=	=	PUNCT
ajird-398	45	3	𝜋	𝜋	X
ajird-398	45	4	→	→	SYM
ajird-398	45	5	𝑡	𝑡	X
ajird-398	45	6	=	=	SYM
ajird-398	45	7	𝜋	𝜋	NOUN
ajird-398	45	8	2	2	NUM
ajird-398	45	9	]	]	PUNCT
ajird-398	45	10	=	=	SYM
ajird-398	45	11	american	american	PROPN
ajird-398	45	12	journal	journal	PROPN
ajird-398	45	13	of	of	ADP
ajird-398	45	14	interdisciplinary	interdisciplinary	ADJ
ajird-398	45	15	research	research	NOUN
ajird-398	45	16	and	and	CCONJ
ajird-398	45	17	development	development	NOUN
ajird-398	45	18	issn	issn	PROPN
ajird-398	45	19	online	online	NOUN
ajird-398	45	20	:	:	PUNCT
ajird-398	45	21	2771	2771	NUM
ajird-398	45	22	-	-	SYM
ajird-398	45	23	8948	8948	NUM
ajird-398	45	24	website	website	NOUN
ajird-398	45	25	:	:	PUNCT
ajird-398	45	26	www.ajird.journalspark.org	www.ajird.journalspark.org	ADJ
ajird-398	45	27	volume	volume	NOUN
ajird-398	45	28	11	11	NUM
ajird-398	45	29	,	,	PUNCT
ajird-398	45	30	dec	dec	PROPN
ajird-398	45	31	.	.	PROPN
ajird-398	45	32	,	,	PUNCT
ajird-398	45	33	2022	2022	NUM
ajird-398	45	34	37	37	NUM
ajird-398	46	1	|	|	ADV
ajird-398	46	2	p	p	NOUN
ajird-398	46	3	a	a	DET
ajird-398	46	4	g	g	NOUN
ajird-398	46	5	e	e	NOUN
ajird-398	46	6	=	=	SYM
ajird-398	47	1	∫	∫	PROPN
ajird-398	48	1	𝑒𝑚𝑟∙cos𝑥	𝑒𝑚𝑟∙cos𝑥	NOUN
ajird-398	48	2	𝜋	𝜋	NOUN
ajird-398	48	3	2	2	NUM
ajird-398	48	4	0	0	NUM
ajird-398	48	5	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	48	6	+	+	ADJ
ajird-398	48	7	∫	∫	PROPN
ajird-398	48	8	𝑒𝑚𝑟∙cos(𝑡+	𝑒𝑚𝑟∙cos(𝑡+	NOUN
ajird-398	48	9	𝜋	𝜋	PRON
ajird-398	48	10	2	2	NUM
ajird-398	48	11	)	)	PUNCT
ajird-398	48	12	𝜋	𝜋	NOUN
ajird-398	48	13	2	2	NUM
ajird-398	48	14	0	0	NUM
ajird-398	48	15	𝑑𝑡	𝑑𝑡	ADP
ajird-398	48	16	=	=	SYM
ajird-398	48	17	∫	∫	PROPN
ajird-398	49	1	𝑒𝑚𝑟∙cos𝑥	𝑒𝑚𝑟∙cos𝑥	NOUN
ajird-398	49	2	𝜋	𝜋	NOUN
ajird-398	49	3	2	2	NUM
ajird-398	49	4	0	0	NUM
ajird-398	49	5	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	49	6	+	+	NUM
ajird-398	49	7	∫	∫	PROPN
ajird-398	49	8	𝑒−𝑚𝑟∙sin𝑥	𝑒−𝑚𝑟∙sin𝑥	NOUN
ajird-398	49	9	𝜋	𝜋	VERB
ajird-398	49	10	2	2	NUM
ajird-398	49	11	0	0	NUM
ajird-398	49	12	𝑑𝑥.	𝑑𝑥.	NOUN
ajird-398	49	13	we	we	PRON
ajird-398	49	14	can	can	AUX
ajird-398	49	15	substitute	substitute	VERB
ajird-398	49	16	in	in	ADP
ajird-398	49	17	the	the	DET
ajird-398	49	18	integrals	integral	NOUN
ajird-398	49	19	on	on	ADP
ajird-398	49	20	the	the	DET
ajird-398	49	21	right	right	ADJ
ajird-398	49	22	side	side	NOUN
ajird-398	49	23	of	of	ADP
ajird-398	49	24	the	the	DET
ajird-398	49	25	last	last	ADJ
ajird-398	49	26	equality	equality	NOUN
ajird-398	49	27	:	:	PUNCT
ajird-398	49	28	∫	∫	PROPN
ajird-398	50	1	𝑒𝑚𝑟∙cos𝑥	𝑒𝑚𝑟∙cos𝑥	NOUN
ajird-398	50	2	𝜋	𝜋	NOUN
ajird-398	50	3	2	2	NUM
ajird-398	50	4	0	0	NUM
ajird-398	50	5	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	50	6	=	=	SYM
ajird-398	51	1	[	[	X
ajird-398	51	2	cos	cos	X
ajird-398	51	3	𝑥	𝑥	NOUN
ajird-398	51	4	=	=	SYM
ajird-398	51	5	𝑡	𝑡	X
ajird-398	51	6	]	]	PUNCT
ajird-398	51	7	=	=	SYM
ajird-398	51	8	∫	∫	PROPN
ajird-398	51	9	𝑒𝑚𝑟𝑡	𝑒𝑚𝑟𝑡	NOUN
ajird-398	51	10	√1	√1	PART
ajird-398	51	11	−	−	PROPN
ajird-398	51	12	𝑡2	𝑡2	NOUN
ajird-398	51	13	𝑑𝑡	𝑑𝑡	ADP
ajird-398	51	14	1	1	NUM
ajird-398	51	15	0	0	NUM
ajird-398	51	16	,	,	PUNCT
ajird-398	51	17	∫	∫	PROPN
ajird-398	51	18	𝑒−𝑚𝑟∙sin𝑥	𝑒−𝑚𝑟∙sin𝑥	PROPN
ajird-398	51	19	𝜋	𝜋	VERB
ajird-398	51	20	2	2	NUM
ajird-398	51	21	0	0	NUM
ajird-398	51	22	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	51	23	=	=	SYM
ajird-398	52	1	[	[	X
ajird-398	52	2	sin	sin	X
ajird-398	52	3	𝑥	𝑥	NOUN
ajird-398	52	4	=	=	PUNCT
ajird-398	52	5	𝑧	𝑧	NOUN
ajird-398	52	6	]	]	X
ajird-398	52	7	=	=	SYM
ajird-398	52	8	∫	∫	PROPN
ajird-398	52	9	𝑒−𝑚𝑟𝑧	𝑒−𝑚𝑟𝑧	NOUN
ajird-398	52	10	√1	√1	ADV
ajird-398	52	11	−	−	PROPN
ajird-398	52	12	𝑧2	𝑧2	PROPN
ajird-398	52	13	𝑑𝑧	𝑑𝑧	NOUN
ajird-398	52	14	1	1	NUM
ajird-398	52	15	0	0	NUM
ajird-398	52	16	=	=	SYM
ajird-398	52	17	∫	∫	PROPN
ajird-398	52	18	𝑒−𝑚𝑟𝑡	𝑒−𝑚𝑟𝑡	NOUN
ajird-398	52	19	√1	√1	PART
ajird-398	53	1	−	−	PROPN
ajird-398	53	2	𝑡2	𝑡2	NOUN
ajird-398	53	3	𝑑𝑡	𝑑𝑡	ADP
ajird-398	53	4	1	1	NUM
ajird-398	53	5	0	0	NUM
ajird-398	53	6	.	.	PUNCT
ajird-398	54	1	then	then	ADV
ajird-398	54	2	,	,	PUNCT
ajird-398	54	3	𝐼(𝑟	𝐼(𝑟	ADV
ajird-398	54	4	)	)	PUNCT
ajird-398	54	5	=	=	SYM
ajird-398	54	6	∫	∫	PROPN
ajird-398	54	7	𝑒𝑚𝑟𝑡	𝑒𝑚𝑟𝑡	NOUN
ajird-398	54	8	+	+	CCONJ
ajird-398	54	9	𝑒−𝑚𝑟𝑡	𝑒−𝑚𝑟𝑡	NOUN
ajird-398	54	10	√1	√1	PART
ajird-398	54	11	−	−	PROPN
ajird-398	54	12	𝑡2	𝑡2	NOUN
ajird-398	54	13	𝑑𝑡	𝑑𝑡	ADP
ajird-398	54	14	1	1	NUM
ajird-398	54	15	0	0	NUM
ajird-398	54	16	.	.	PUNCT
ajird-398	55	1	now	now	ADV
ajird-398	55	2	we	we	PRON
ajird-398	55	3	use	use	VERB
ajird-398	55	4	this	this	DET
ajird-398	55	5	expansion	expansion	NOUN
ajird-398	55	6	:	:	PUNCT
ajird-398	55	7	𝑒𝑥	𝑒𝑥	PROPN
ajird-398	55	8	=	=	PUNCT
ajird-398	55	9	∑	∑	PROPN
ajird-398	55	10	𝑥𝑛	𝑥𝑛	PROPN
ajird-398	55	11	𝑛	𝑛	PROPN
ajird-398	55	12	!	!	PUNCT
ajird-398	55	13	∞	∞	NUM
ajird-398	55	14	𝑛=0	𝑛=0	PROPN
ajird-398	55	15	.	.	PUNCT
ajird-398	56	1	then	then	ADV
ajird-398	56	2	we	we	PRON
ajird-398	56	3	arrive	arrive	VERB
ajird-398	56	4	at	at	ADP
ajird-398	56	5	the	the	DET
ajird-398	56	6	following	follow	VERB
ajird-398	56	7	equality	equality	NOUN
ajird-398	56	8	:	:	PUNCT
ajird-398	56	9	𝐼(𝑟	𝐼(𝑟	NUM
ajird-398	56	10	)	)	PUNCT
ajird-398	56	11	=	=	SYM
ajird-398	56	12	∫	∫	PROPN
ajird-398	56	13	(	(	PUNCT
ajird-398	56	14	1	1	NUM
ajird-398	56	15	√1	√1	PROPN
ajird-398	56	16	−	−	PROPN
ajird-398	56	17	𝑡2	𝑡2	PROPN
ajird-398	56	18	∑	∑	PUNCT
ajird-398	56	19	(	(	PUNCT
ajird-398	56	20	𝑚𝑟𝑡)𝑛	𝑚𝑟𝑡)𝑛	PROPN
ajird-398	56	21	+	+	NUM
ajird-398	56	22	(	(	PUNCT
ajird-398	56	23	−𝑚𝑟𝑡)𝑛	−𝑚𝑟𝑡)𝑛	PROPN
ajird-398	56	24	𝑛	𝑛	PROPN
ajird-398	56	25	!	!	PUNCT
ajird-398	56	26	∞	∞	NUM
ajird-398	56	27	𝑛=0	𝑛=0	PROPN
ajird-398	56	28	)	)	PUNCT
ajird-398	56	29	1	1	NUM
ajird-398	56	30	0	0	NUM
ajird-398	56	31	𝑑𝑡.	𝑑𝑡.	NOUN
ajird-398	56	32	it	it	PRON
ajird-398	56	33	is	be	AUX
ajird-398	56	34	known	know	VERB
ajird-398	56	35	that	that	SCONJ
ajird-398	56	36	in	in	ADP
ajird-398	56	37	the	the	DET
ajird-398	56	38	sum	sum	NOUN
ajird-398	56	39	on	on	ADP
ajird-398	56	40	the	the	DET
ajird-398	56	41	right	right	ADJ
ajird-398	56	42	side	side	NOUN
ajird-398	56	43	of	of	ADP
ajird-398	56	44	the	the	DET
ajird-398	56	45	above	above	ADJ
ajird-398	56	46	equation	equation	NOUN
ajird-398	56	47	,	,	PUNCT
ajird-398	56	48	the	the	DET
ajird-398	56	49	odd	odd	ADV
ajird-398	56	50	-	-	PUNCT
ajird-398	56	51	numbered	number	VERB
ajird-398	56	52	terms	term	NOUN
ajird-398	56	53	become	become	VERB
ajird-398	56	54	zero	zero	NUM
ajird-398	56	55	.	.	PUNCT
ajird-398	57	1	so	so	ADV
ajird-398	57	2	,	,	PUNCT
ajird-398	57	3	𝐼(𝑟	𝐼(𝑟	NOUN
ajird-398	57	4	)	)	PUNCT
ajird-398	57	5	=	=	SYM
ajird-398	57	6	2∫	2∫	NUM
ajird-398	57	7	∑	∑	INTJ
ajird-398	57	8	(	(	PUNCT
ajird-398	57	9	𝑚	𝑚	ADP
ajird-398	57	10	∙	∙	PROPN
ajird-398	57	11	𝑟)2𝑛𝑡2𝑛	𝑟)2𝑛𝑡2𝑛	NOUN
ajird-398	57	12	2𝑛	2𝑛	NUM
ajird-398	57	13	!	!	PUNCT
ajird-398	58	1	∙	∙	PROPN
ajird-398	58	2	√1	√1	PART
ajird-398	59	1	−	−	PROPN
ajird-398	59	2	𝑡2	𝑡2	NOUN
ajird-398	59	3	∞	∞	NUM
ajird-398	59	4	𝑛=0	𝑛=0	PROPN
ajird-398	59	5	1	1	NUM
ajird-398	59	6	0	0	NUM
ajird-398	59	7	𝑑𝑡	𝑑𝑡	ADP
ajird-398	59	8	=	=	PRON
ajird-398	59	9	∑	∑	PROPN
ajird-398	59	10	(	(	PUNCT
ajird-398	59	11	𝑚𝑟)2𝑛	𝑚𝑟)2𝑛	NOUN
ajird-398	59	12	(	(	PUNCT
ajird-398	59	13	2𝑛	2𝑛	NUM
ajird-398	59	14	)	)	PUNCT
ajird-398	59	15	!	!	PUNCT
ajird-398	60	1	∫	∫	PROPN
ajird-398	61	1	2𝑡2𝑛	2𝑡2𝑛	NOUN
ajird-398	61	2	√1	√1	PROPN
ajird-398	62	1	−	−	PROPN
ajird-398	62	2	𝑡2	𝑡2	PROPN
ajird-398	62	3	1	1	NUM
ajird-398	62	4	0	0	NUM
ajird-398	62	5	𝑑𝑡	𝑑𝑡	ADP
ajird-398	62	6	∞	∞	NUM
ajird-398	62	7	𝑛=0	𝑛=0	PROPN
ajird-398	62	8	=	=	PUNCT
ajird-398	63	1	[	[	PUNCT
ajird-398	63	2	𝑡2	𝑡2	NOUN
ajird-398	63	3	=	=	PUNCT
ajird-398	63	4	𝑥	𝑥	X
ajird-398	63	5	𝑡	𝑡	PROPN
ajird-398	63	6	=	=	NOUN
ajird-398	63	7	√𝑥	√𝑥	NOUN
ajird-398	63	8	𝑑𝑡	𝑑𝑡	ADP
ajird-398	63	9	=	=	SYM
ajird-398	63	10	1	1	NUM
ajird-398	63	11	2√𝑥	2√𝑥	NOUN
ajird-398	63	12	𝑑𝑥	𝑑𝑥	VERB
ajird-398	63	13	]	]	PUNCT
ajird-398	63	14	=	=	SYM
ajird-398	63	15	=	=	SYM
ajird-398	63	16	2	2	NUM
ajird-398	63	17	∑	∑	PUNCT
ajird-398	63	18	(	(	PUNCT
ajird-398	63	19	𝑚𝑟)2𝑛	𝑚𝑟)2𝑛	NOUN
ajird-398	63	20	(	(	PUNCT
ajird-398	63	21	2𝑛	2𝑛	NUM
ajird-398	63	22	)	)	PUNCT
ajird-398	63	23	!	!	PUNCT
ajird-398	64	1	∫	∫	PROPN
ajird-398	65	1	𝑥𝑛	𝑥𝑛	PROPN
ajird-398	65	2	(	(	PUNCT
ajird-398	65	3	1	1	NUM
ajird-398	65	4	−	−	PROPN
ajird-398	65	5	𝑥	𝑥	NOUN
ajird-398	65	6	)	)	PUNCT
ajird-398	65	7	1	1	NUM
ajird-398	65	8	2	2	NUM
ajird-398	65	9	∙	∙	NOUN
ajird-398	65	10	1	1	NUM
ajird-398	65	11	2√𝑥	2√𝑥	NUM
ajird-398	65	12	1	1	NUM
ajird-398	65	13	0	0	NUM
ajird-398	65	14	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	65	15	∞	∞	NUM
ajird-398	65	16	𝑛=0	𝑛=0	PROPN
ajird-398	65	17	=	=	PUNCT
ajird-398	65	18	∑	∑	PUNCT
ajird-398	65	19	(	(	PUNCT
ajird-398	65	20	𝑚𝑟)2𝑛	𝑚𝑟)2𝑛	NOUN
ajird-398	65	21	(	(	PUNCT
ajird-398	65	22	2𝑛	2𝑛	NUM
ajird-398	65	23	)	)	PUNCT
ajird-398	65	24	!	!	PUNCT
ajird-398	66	1	∫	∫	PROPN
ajird-398	66	2	𝑥𝑛−	𝑥𝑛−	NUM
ajird-398	66	3	1	1	NUM
ajird-398	66	4	2	2	NUM
ajird-398	66	5	∙	∙	X
ajird-398	66	6	(	(	PUNCT
ajird-398	66	7	1	1	NUM
ajird-398	66	8	−	−	NOUN
ajird-398	66	9	𝑥)−	𝑥)−	PROPN
ajird-398	66	10	1	1	NUM
ajird-398	67	1	2	2	NUM
ajird-398	67	2	1	1	NUM
ajird-398	67	3	0	0	NUM
ajird-398	67	4	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	67	5	∞	∞	NUM
ajird-398	67	6	𝑛=0	𝑛=0	PROPN
ajird-398	67	7	.	.	PUNCT
ajird-398	68	1	the	the	DET
ajird-398	68	2	integral	integral	ADJ
ajird-398	68	3	on	on	ADP
ajird-398	68	4	the	the	DET
ajird-398	68	5	right	right	ADJ
ajird-398	68	6	side	side	NOUN
ajird-398	68	7	of	of	ADP
ajird-398	68	8	the	the	DET
ajird-398	68	9	resulting	result	VERB
ajird-398	68	10	equation	equation	NOUN
ajird-398	68	11	can	can	AUX
ajird-398	68	12	be	be	AUX
ajird-398	68	13	calculated	calculate	VERB
ajird-398	68	14	using	use	VERB
ajird-398	68	15	the	the	DET
ajird-398	68	16	beta	beta	ADJ
ajird-398	68	17	function	function	NOUN
ajird-398	68	18	:	:	PUNCT
ajird-398	68	19	𝐼(𝑟	𝐼(𝑟	X
ajird-398	68	20	)	)	PUNCT
ajird-398	68	21	=	=	PUNCT
ajird-398	69	1	∑	∑	PUNCT
ajird-398	69	2	(	(	PUNCT
ajird-398	69	3	𝑚𝑟)2𝑛	𝑚𝑟)2𝑛	NOUN
ajird-398	69	4	(	(	PUNCT
ajird-398	69	5	2𝑛	2𝑛	NUM
ajird-398	69	6	)	)	PUNCT
ajird-398	69	7	!	!	PUNCT
ajird-398	70	1	∙	∙	PROPN
ajird-398	70	2	𝐵	𝐵	PROPN
ajird-398	70	3	(	(	PUNCT
ajird-398	70	4	𝑛	𝑛	PROPN
ajird-398	70	5	+	+	NUM
ajird-398	70	6	1	1	NUM
ajird-398	70	7	2	2	NUM
ajird-398	70	8	;	;	PUNCT
ajird-398	70	9	1	1	NUM
ajird-398	70	10	2	2	NUM
ajird-398	70	11	)	)	PUNCT
ajird-398	70	12	∞	∞	NUM
ajird-398	70	13	𝑛=0	𝑛=0	PROPN
ajird-398	70	14	=	=	PUNCT
ajird-398	70	15	∑	∑	PUNCT
ajird-398	70	16	(	(	PUNCT
ajird-398	70	17	𝑚𝑟)2𝑛	𝑚𝑟)2𝑛	NOUN
ajird-398	70	18	(	(	PUNCT
ajird-398	70	19	2𝑛	2𝑛	NUM
ajird-398	70	20	)	)	PUNCT
ajird-398	70	21	!	!	PUNCT
ajird-398	71	1	∙	∙	PROPN
ajird-398	71	2	г	г	PROPN
ajird-398	71	3	(	(	PUNCT
ajird-398	71	4	𝑛	𝑛	PROPN
ajird-398	71	5	+	+	NUM
ajird-398	71	6	1	1	NUM
ajird-398	71	7	2	2	NUM
ajird-398	71	8	)	)	PUNCT
ajird-398	71	9	∙	∙	PROPN
ajird-398	71	10	г	г	PROPN
ajird-398	71	11	(	(	PUNCT
ajird-398	71	12	1	1	NUM
ajird-398	71	13	2	2	NUM
ajird-398	71	14	)	)	PUNCT
ajird-398	71	15	г(𝑛	г(𝑛	NOUN
ajird-398	72	1	+	+	CCONJ
ajird-398	72	2	1	1	X
ajird-398	72	3	)	)	PUNCT
ajird-398	72	4	∞	∞	NUM
ajird-398	72	5	𝑛=0	𝑛=0	X
ajird-398	72	6	=	=	PUNCT
ajird-398	73	1	=	=	PUNCT
ajird-398	73	2	∑	∑	PUNCT
ajird-398	73	3	𝑚2𝑛𝑟2𝑛	𝑚2𝑛𝑟2𝑛	PUNCT
ajird-398	73	4	∙	∙	PROPN
ajird-398	73	5	𝜋(2𝑛	𝜋(2𝑛	NUM
ajird-398	73	6	−	−	PROPN
ajird-398	73	7	1)‼	1)‼	PROPN
ajird-398	73	8	2𝑛	2𝑛	PROPN
ajird-398	73	9	∙	∙	PROPN
ajird-398	73	10	𝑛	𝑛	PROPN
ajird-398	73	11	!	!	PUNCT
ajird-398	74	1	∙	∙	PROPN
ajird-398	74	2	(	(	PUNCT
ajird-398	74	3	2𝑛	2𝑛	NUM
ajird-398	74	4	)	)	PUNCT
ajird-398	74	5	!	!	PUNCT
ajird-398	75	1	∞	∞	NUM
ajird-398	76	1	𝑛=0	𝑛=0	X
ajird-398	76	2	=	=	PUNCT
ajird-398	76	3	∑	∑	PUNCT
ajird-398	76	4	𝑚2𝑛𝑟2𝑛𝜋	𝑚2𝑛𝑟2𝑛𝜋	NOUN
ajird-398	76	5	22𝑛(𝑛!)2	22𝑛(𝑛!)2	NUM
ajird-398	76	6	∞	∞	NUM
ajird-398	76	7	𝑛=0	𝑛=0	PROPN
ajird-398	76	8	.	.	PUNCT
ajird-398	77	1	therefore	therefore	ADV
ajird-398	77	2	,	,	PUNCT
ajird-398	77	3	𝐼(𝑟	𝐼(𝑟	ADV
ajird-398	77	4	)	)	PUNCT
ajird-398	77	5	=	=	PUNCT
ajird-398	77	6	∑	∑	PUNCT
ajird-398	77	7	𝑚2𝑛𝑟2𝑛𝜋	𝑚2𝑛𝑟2𝑛𝜋	NOUN
ajird-398	77	8	22𝑛(𝑛!)2	22𝑛(𝑛!)2	NUM
ajird-398	77	9	∞	∞	NUM
ajird-398	77	10	𝑛=0	𝑛=0	PROPN
ajird-398	77	11	.	.	PUNCT
ajird-398	78	1	problem	problem	NOUN
ajird-398	78	2	2	2	NUM
ajird-398	78	3	.	.	PUNCT
ajird-398	78	4	evaluate	evaluate	VERB
ajird-398	78	5	the	the	DET
ajird-398	78	6	value	value	NOUN
ajird-398	78	7	of	of	ADP
ajird-398	78	8	the	the	DET
ajird-398	78	9	following	follow	VERB
ajird-398	78	10	integral	integral	ADJ
ajird-398	78	11	:	:	PUNCT
ajird-398	79	1	𝐼(𝑏	𝐼(𝑏	X
ajird-398	79	2	)	)	PUNCT
ajird-398	79	3	=	=	SYM
ajird-398	79	4	∫	∫	PROPN
ajird-398	79	5	𝑒−𝑎𝑥2	𝑒−𝑎𝑥2	PROPN
ajird-398	79	6	cos	cos	PROPN
ajird-398	79	7	𝑏𝑥	𝑏𝑥	PROPN
ajird-398	80	1	+	+	PROPN
ajird-398	80	2	∞	∞	PROPN
ajird-398	80	3	0	0	NUM
ajird-398	80	4	𝑑𝑥.	𝑑𝑥.	PROPN
ajird-398	80	5	american	american	PROPN
ajird-398	80	6	journal	journal	PROPN
ajird-398	80	7	of	of	ADP
ajird-398	80	8	interdisciplinary	interdisciplinary	ADJ
ajird-398	80	9	research	research	NOUN
ajird-398	80	10	and	and	CCONJ
ajird-398	80	11	development	development	NOUN
ajird-398	80	12	issn	issn	PROPN
ajird-398	80	13	online	online	NOUN
ajird-398	80	14	:	:	PUNCT
ajird-398	80	15	2771	2771	NUM
ajird-398	80	16	-	-	SYM
ajird-398	80	17	8948	8948	NUM
ajird-398	80	18	website	website	NOUN
ajird-398	80	19	:	:	PUNCT
ajird-398	80	20	www.ajird.journalspark.org	www.ajird.journalspark.org	ADJ
ajird-398	80	21	volume	volume	NOUN
ajird-398	80	22	11	11	NUM
ajird-398	80	23	,	,	PUNCT
ajird-398	80	24	dec	dec	PROPN
ajird-398	80	25	.	.	PROPN
ajird-398	80	26	,	,	PUNCT
ajird-398	80	27	2022	2022	NUM
ajird-398	80	28	38	38	NUM
ajird-398	81	1	|	|	ADV
ajird-398	81	2	p	p	NOUN
ajird-398	81	3	a	a	DET
ajird-398	81	4	g	g	NOUN
ajird-398	81	5	e	e	NOUN
ajird-398	81	6	solution	solution	NOUN
ajird-398	81	7	.	.	PUNCT
ajird-398	82	1	it	it	PRON
ajird-398	82	2	is	be	AUX
ajird-398	82	3	known	know	VERB
ajird-398	82	4	that	that	SCONJ
ajird-398	82	5	the	the	DET
ajird-398	82	6	given	give	VERB
ajird-398	82	7	integral	integral	ADJ
ajird-398	82	8	depending	depending	NOUN
ajird-398	82	9	on	on	ADP
ajird-398	82	10	the	the	DET
ajird-398	82	11	parameter	parameter	NOUN
ajird-398	82	12	is	be	AUX
ajird-398	82	13	uniformly	uniformly	ADV
ajird-398	82	14	convergent	convergent	NOUN
ajird-398	82	15	according	accord	VERB
ajird-398	82	16	to	to	ADP
ajird-398	82	17	the	the	DET
ajird-398	82	18	comparison	comparison	NOUN
ajird-398	82	19	property	property	NOUN
ajird-398	82	20	.	.	PUNCT
ajird-398	83	1	so	so	ADV
ajird-398	83	2	,	,	PUNCT
ajird-398	83	3	the	the	DET
ajird-398	83	4	sign	sign	NOUN
ajird-398	83	5	of	of	ADP
ajird-398	83	6	integral	integral	ADJ
ajird-398	83	7	can	can	AUX
ajird-398	83	8	be	be	AUX
ajird-398	83	9	replaced	replace	VERB
ajird-398	83	10	by	by	ADP
ajird-398	83	11	the	the	DET
ajird-398	83	12	sign	sign	NOUN
ajird-398	83	13	of	of	ADP
ajird-398	83	14	sum	sum	NOUN
ajird-398	83	15	.	.	PUNCT
ajird-398	84	1	to	to	PART
ajird-398	84	2	calculate	calculate	VERB
ajird-398	84	3	the	the	DET
ajird-398	84	4	value	value	NOUN
ajird-398	84	5	of	of	ADP
ajird-398	84	6	this	this	DET
ajird-398	84	7	integral	integral	ADJ
ajird-398	84	8	,	,	PUNCT
ajird-398	84	9	we	we	PRON
ajird-398	84	10	use	use	VERB
ajird-398	84	11	the	the	DET
ajird-398	84	12	following	follow	VERB
ajird-398	84	13	taylor	taylor	PROPN
ajird-398	84	14	expansion	expansion	NOUN
ajird-398	84	15	:	:	PUNCT
ajird-398	85	1	cos	cos	PROPN
ajird-398	85	2	𝑏𝑥	𝑏𝑥	PROPN
ajird-398	85	3	=	=	SYM
ajird-398	85	4	∑	∑	PROPN
ajird-398	85	5	(	(	PUNCT
ajird-398	85	6	−1)𝑛(𝑏𝑥)2𝑛	−1)𝑛(𝑏𝑥)2𝑛	PROPN
ajird-398	85	7	(	(	PUNCT
ajird-398	85	8	2𝑛	2𝑛	NUM
ajird-398	85	9	)	)	PUNCT
ajird-398	85	10	!	!	PUNCT
ajird-398	86	1	∞	∞	NUM
ajird-398	86	2	𝑛=0	𝑛=0	PROPN
ajird-398	86	3	.	.	PUNCT
ajird-398	87	1	in	in	ADP
ajird-398	87	2	that	that	DET
ajird-398	87	3	case	case	NOUN
ajird-398	87	4	,	,	PUNCT
ajird-398	87	5	𝐼(𝑏	𝐼(𝑏	NOUN
ajird-398	87	6	)	)	PUNCT
ajird-398	87	7	=	=	SYM
ajird-398	87	8	∫	∫	PROPN
ajird-398	87	9	𝑒−𝑎𝑥2	𝑒−𝑎𝑥2	INTJ
ajird-398	87	10	∑	∑	PROPN
ajird-398	87	11	(	(	PUNCT
ajird-398	87	12	−1)𝑛(𝑏𝑥)2𝑛	−1)𝑛(𝑏𝑥)2𝑛	PROPN
ajird-398	87	13	(	(	PUNCT
ajird-398	87	14	2𝑛	2𝑛	NUM
ajird-398	87	15	)	)	PUNCT
ajird-398	87	16	!	!	PUNCT
ajird-398	88	1	∞	∞	NUM
ajird-398	89	1	𝑛=0	𝑛=0	X
ajird-398	90	1	+	+	NOUN
ajird-398	90	2	∞	∞	NOUN
ajird-398	90	3	0	0	NUM
ajird-398	90	4	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	90	5	=	=	SYM
ajird-398	90	6	∑	∑	NOUN
ajird-398	90	7	∫	∫	PROPN
ajird-398	90	8	𝑒−𝑎𝑥2	𝑒−𝑎𝑥2	PROPN
ajird-398	90	9	(	(	PUNCT
ajird-398	90	10	−1)𝑛(𝑏𝑥)2𝑛	−1)𝑛(𝑏𝑥)2𝑛	PROPN
ajird-398	90	11	(	(	PUNCT
ajird-398	90	12	2𝑛	2𝑛	NUM
ajird-398	90	13	)	)	PUNCT
ajird-398	90	14	!	!	PUNCT
ajird-398	91	1	+	+	PUNCT
ajird-398	91	2	∞	∞	NOUN
ajird-398	91	3	0	0	NUM
ajird-398	91	4	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	91	5	∞	∞	NUM
ajird-398	91	6	𝑛=0	𝑛=0	PROPN
ajird-398	91	7	=	=	PUNCT
ajird-398	92	1	=	=	PUNCT
ajird-398	92	2	[	[	PUNCT
ajird-398	92	3	𝑎𝑥2	𝑎𝑥2	X
ajird-398	92	4	=	=	SYM
ajird-398	92	5	𝑡	𝑡	PART
ajird-398	92	6	𝑥	𝑥	NOUN
ajird-398	92	7	=	=	PUNCT
ajird-398	92	8	√	√	NUM
ajird-398	92	9	𝑡	𝑡	NOUN
ajird-398	92	10	𝑎	𝑎	PRON
ajird-398	92	11	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	92	12	=	=	SYM
ajird-398	92	13	1	1	NUM
ajird-398	92	14	2√𝑎𝑡	2√𝑎𝑡	NUM
ajird-398	92	15	𝑑𝑡	𝑑𝑡	ADP
ajird-398	92	16	]	]	PUNCT
ajird-398	92	17	=	=	PUNCT
ajird-398	92	18	∑	∑	PUNCT
ajird-398	92	19	(	(	PUNCT
ajird-398	92	20	−1)𝑛𝑏2𝑛	−1)𝑛𝑏2𝑛	PROPN
ajird-398	92	21	(	(	PUNCT
ajird-398	92	22	2𝑛	2𝑛	NUM
ajird-398	92	23	)	)	PUNCT
ajird-398	92	24	!	!	PUNCT
ajird-398	93	1	∫	∫	PROPN
ajird-398	94	1	𝑒−𝑡	𝑒−𝑡	PROPN
ajird-398	94	2	(	(	PUNCT
ajird-398	94	3	√	√	PROPN
ajird-398	94	4	𝑡	𝑡	PRON
ajird-398	94	5	𝑎	𝑎	NOUN
ajird-398	94	6	)	)	PUNCT
ajird-398	94	7	2𝑛	2𝑛	PROPN
ajird-398	94	8	(	(	PUNCT
ajird-398	94	9	1	1	NUM
ajird-398	94	10	2√𝑎𝑡	2√𝑎𝑡	NUM
ajird-398	94	11	)	)	PUNCT
ajird-398	95	1	+	+	ADP
ajird-398	95	2	∞	∞	NOUN
ajird-398	95	3	0	0	NUM
ajird-398	95	4	𝑑𝑡	𝑑𝑡	ADP
ajird-398	95	5	∞	∞	NUM
ajird-398	95	6	𝑛=0	𝑛=0	PROPN
ajird-398	95	7	=	=	PUNCT
ajird-398	95	8	=	=	PUNCT
ajird-398	95	9	∑	∑	PUNCT
ajird-398	95	10	(	(	PUNCT
ajird-398	95	11	−1)𝑛𝑏2𝑛	−1)𝑛𝑏2𝑛	NOUN
ajird-398	95	12	2(2𝑛	2(2𝑛	NUM
ajird-398	95	13	)	)	PUNCT
ajird-398	95	14	!	!	PUNCT
ajird-398	96	1	𝑎𝑛+	𝑎𝑛+	NOUN
ajird-398	96	2	1	1	NUM
ajird-398	96	3	2	2	NUM
ajird-398	96	4	∫	∫	NOUN
ajird-398	96	5	𝑒−𝑡𝑡𝑛−	𝑒−𝑡𝑡𝑛−	NOUN
ajird-398	96	6	1	1	NUM
ajird-398	96	7	2	2	NUM
ajird-398	96	8	+	+	NOUN
ajird-398	96	9	∞	∞	NOUN
ajird-398	96	10	0	0	NUM
ajird-398	96	11	𝑑𝑡	𝑑𝑡	ADP
ajird-398	96	12	∞	∞	NUM
ajird-398	96	13	𝑛=0	𝑛=0	PROPN
ajird-398	96	14	=	=	PUNCT
ajird-398	96	15	∑	∑	PUNCT
ajird-398	96	16	(	(	PUNCT
ajird-398	96	17	−1)𝑛𝑏2𝑛	−1)𝑛𝑏2𝑛	NOUN
ajird-398	96	18	2(2𝑛	2(2𝑛	NUM
ajird-398	96	19	)	)	PUNCT
ajird-398	96	20	!	!	PUNCT
ajird-398	97	1	𝑎𝑛+	𝑎𝑛+	NOUN
ajird-398	97	2	1	1	NUM
ajird-398	97	3	2	2	NUM
ajird-398	97	4	г	г	PROPN
ajird-398	97	5	(	(	PUNCT
ajird-398	97	6	𝑛	𝑛	PROPN
ajird-398	97	7	+	+	NOUN
ajird-398	97	8	1	1	NUM
ajird-398	97	9	2	2	NUM
ajird-398	97	10	)	)	PUNCT
ajird-398	97	11	∞	∞	NUM
ajird-398	98	1	𝑛=0	𝑛=0	X
ajird-398	99	1	=	=	PUNCT
ajird-398	100	1	=	=	SYM
ajird-398	101	1	1	1	NUM
ajird-398	101	2	2	2	NUM
ajird-398	101	3	∑	∑	PUNCT
ajird-398	101	4	(	(	PUNCT
ajird-398	101	5	−1)𝑛𝑏2𝑛	−1)𝑛𝑏2𝑛	NOUN
ajird-398	101	6	2(2𝑛	2(2𝑛	NUM
ajird-398	101	7	)	)	PUNCT
ajird-398	101	8	!	!	PUNCT
ajird-398	102	1	𝑎𝑛+	𝑎𝑛+	NOUN
ajird-398	102	2	1	1	NUM
ajird-398	102	3	2	2	NUM
ajird-398	102	4	∙	∙	X
ajird-398	102	5	(	(	PUNCT
ajird-398	102	6	2𝑛	2𝑛	NOUN
ajird-398	102	7	−	−	PROPN
ajird-398	102	8	1)‼√𝜋	1)‼√𝜋	NUM
ajird-398	102	9	2𝑛	2𝑛	NOUN
ajird-398	103	1	∞	∞	NUM
ajird-398	103	2	𝑛=0	𝑛=0	X
ajird-398	103	3	=	=	SYM
ajird-398	103	4	1	1	NUM
ajird-398	103	5	2	2	NUM
ajird-398	103	6	√	√	NUM
ajird-398	103	7	𝜋	𝜋	PRON
ajird-398	103	8	𝛼	𝛼	NOUN
ajird-398	103	9	∑	∑	PUNCT
ajird-398	103	10	(	(	PUNCT
ajird-398	103	11	−1)𝑛𝑏2𝑛	−1)𝑛𝑏2𝑛	NOUN
ajird-398	103	12	2𝑛	2𝑛	PROPN
ajird-398	103	13	∙	∙	PROPN
ajird-398	103	14	(	(	PUNCT
ajird-398	103	15	2𝑛)‼𝑎𝑛	2𝑛)‼𝑎𝑛	NUM
ajird-398	103	16	∞	∞	NOUN
ajird-398	103	17	𝑛=0	𝑛=0	NOUN
ajird-398	103	18	=	=	SYM
ajird-398	103	19	1	1	NUM
ajird-398	103	20	2	2	NUM
ajird-398	103	21	√	√	NUM
ajird-398	103	22	𝜋	𝜋	PRON
ajird-398	103	23	𝛼	𝛼	NOUN
ajird-398	103	24	∑	∑	PUNCT
ajird-398	103	25	(	(	PUNCT
ajird-398	103	26	−1)𝑛𝑏2𝑛	−1)𝑛𝑏2𝑛	NOUN
ajird-398	103	27	4𝑛𝑎𝑛	4𝑛𝑎𝑛	PROPN
ajird-398	103	28	∙	∙	PROPN
ajird-398	103	29	𝑛	𝑛	PROPN
ajird-398	103	30	!	!	NOUN
ajird-398	103	31	∞	∞	NUM
ajird-398	103	32	𝑛=0	𝑛=0	PROPN
ajird-398	103	33	.	.	PUNCT
ajird-398	104	1	so	so	ADV
ajird-398	104	2	,	,	PUNCT
ajird-398	104	3	the	the	DET
ajird-398	104	4	following	follow	VERB
ajird-398	104	5	equality	equality	NOUN
ajird-398	104	6	is	be	AUX
ajird-398	104	7	hold	hold	NOUN
ajird-398	104	8	for	for	ADP
ajird-398	104	9	the	the	DET
ajird-398	104	10	given	give	VERB
ajird-398	104	11	integral	integral	ADJ
ajird-398	104	12	:	:	PUNCT
ajird-398	104	13	𝐼(𝑏	𝐼(𝑏	ADJ
ajird-398	104	14	)	)	PUNCT
ajird-398	104	15	=	=	SYM
ajird-398	104	16	1	1	NUM
ajird-398	104	17	2	2	NUM
ajird-398	104	18	√	√	NUM
ajird-398	104	19	𝜋	𝜋	PRON
ajird-398	104	20	𝛼	𝛼	NOUN
ajird-398	104	21	∑	∑	PUNCT
ajird-398	104	22	[	[	PUNCT
ajird-398	104	23	(	(	PUNCT
ajird-398	104	24	−1)𝑛	−1)𝑛	X
ajird-398	104	25	𝑛	𝑛	NOUN
ajird-398	104	26	!	!	PUNCT
ajird-398	105	1	∙	∙	PROPN
ajird-398	105	2	(	(	PUNCT
ajird-398	105	3	𝑏2	𝑏2	PROPN
ajird-398	105	4	4𝑎	4𝑎	PROPN
ajird-398	105	5	)	)	PUNCT
ajird-398	105	6	𝑛	𝑛	X
ajird-398	105	7	]	]	PUNCT
ajird-398	105	8	∞	∞	NUM
ajird-398	105	9	𝑛=0	𝑛=0	PROPN
ajird-398	105	10	.	.	PUNCT
ajird-398	106	1	it	it	PRON
ajird-398	106	2	is	be	AUX
ajird-398	106	3	not	not	PART
ajird-398	106	4	difficult	difficult	ADJ
ajird-398	106	5	to	to	PART
ajird-398	106	6	know	know	VERB
ajird-398	106	7	that	that	SCONJ
ajird-398	106	8	the	the	DET
ajird-398	106	9	right	right	ADJ
ajird-398	106	10	side	side	NOUN
ajird-398	106	11	of	of	ADP
ajird-398	106	12	this	this	DET
ajird-398	106	13	equation	equation	NOUN
ajird-398	106	14	is	be	AUX
ajird-398	106	15	the	the	DET
ajird-398	106	16	taylor	taylor	PROPN
ajird-398	106	17	expansion	expansion	NOUN
ajird-398	106	18	of	of	ADP
ajird-398	106	19	the	the	DET
ajird-398	106	20	function	function	NOUN
ajird-398	106	21	𝑓(𝑏	𝑓(𝑏	NOUN
ajird-398	106	22	)	)	PUNCT
ajird-398	107	1	=	=	PUNCT
ajird-398	107	2	𝑒−	𝑒−	PROPN
ajird-398	107	3	𝑏2	𝑏2	PROPN
ajird-398	107	4	4𝑎.	4𝑎.	PROPN
ajird-398	108	1	so	so	SCONJ
ajird-398	108	2	the	the	DET
ajird-398	108	3	value	value	NOUN
ajird-398	108	4	of	of	ADP
ajird-398	108	5	the	the	DET
ajird-398	108	6	given	give	VERB
ajird-398	108	7	integral	integral	ADJ
ajird-398	108	8	is	be	AUX
ajird-398	108	9	equal	equal	ADJ
ajird-398	108	10	to	to	ADP
ajird-398	108	11	:	:	PUNCT
ajird-398	108	12	𝑢(𝑏	𝑢(𝑏	PROPN
ajird-398	108	13	)	)	PUNCT
ajird-398	109	1	=	=	SYM
ajird-398	109	2	∫	∫	PROPN
ajird-398	109	3	𝑒−𝑎𝑥2	𝑒−𝑎𝑥2	PROPN
ajird-398	109	4	cos	cos	PROPN
ajird-398	109	5	𝑏𝑥	𝑏𝑥	PROPN
ajird-398	110	1	+	+	PROPN
ajird-398	110	2	∞	∞	NOUN
ajird-398	110	3	0	0	NUM
ajird-398	110	4	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	110	5	=	=	SYM
ajird-398	110	6	1	1	NUM
ajird-398	110	7	2	2	NUM
ajird-398	110	8	√	√	NUM
ajird-398	110	9	𝜋	𝜋	PRON
ajird-398	110	10	𝛼	𝛼	NOUN
ajird-398	110	11	∑	∑	PUNCT
ajird-398	110	12	(	(	PUNCT
ajird-398	110	13	−1)𝑛𝑏2𝑛	−1)𝑛𝑏2𝑛	NOUN
ajird-398	110	14	4𝑛𝑎𝑛	4𝑛𝑎𝑛	PROPN
ajird-398	110	15	∙	∙	PROPN
ajird-398	110	16	𝑛	𝑛	PROPN
ajird-398	110	17	!	!	NOUN
ajird-398	110	18	∞	∞	NUM
ajird-398	111	1	𝑛=0	𝑛=0	X
ajird-398	111	2	=	=	SYM
ajird-398	111	3	1	1	NUM
ajird-398	111	4	2	2	NUM
ajird-398	111	5	√	√	NUM
ajird-398	111	6	𝜋	𝜋	NOUN
ajird-398	111	7	𝑎	𝑎	DET
ajird-398	111	8	𝑒−	𝑒−	NOUN
ajird-398	111	9	𝑏2	𝑏2	PROPN
ajird-398	111	10	4𝑎	4𝑎	PROPN
ajird-398	111	11	.	.	PUNCT
ajird-398	112	1	problem	problem	NOUN
ajird-398	112	2	3	3	NUM
ajird-398	112	3	.	.	PUNCT
ajird-398	112	4	evaluate	evaluate	VERB
ajird-398	112	5	the	the	DET
ajird-398	112	6	value	value	NOUN
ajird-398	112	7	of	of	ADP
ajird-398	112	8	the	the	DET
ajird-398	112	9	following	follow	VERB
ajird-398	112	10	integral	integral	ADJ
ajird-398	112	11	:	:	PUNCT
ajird-398	112	12	∫	∫	PROPN
ajird-398	112	13	𝑓(𝑥)(𝑏	𝑓(𝑥)(𝑏	NUM
ajird-398	112	14	−	−	PROPN
ajird-398	112	15	𝑥)𝛼𝑏	𝑥)𝛼𝑏	PROPN
ajird-398	112	16	0	0	NUM
ajird-398	112	17	𝑑𝑥.	𝑑𝑥.	NOUN
ajird-398	112	18	here	here	ADV
ajird-398	112	19	,	,	PUNCT
ajird-398	112	20	𝛼	𝛼	X
ajird-398	112	21	>	>	X
ajird-398	112	22	−1	−1	NOUN
ajird-398	112	23	and	and	CCONJ
ajird-398	112	24	𝑓(𝑥	𝑓(𝑥	NOUN
ajird-398	112	25	)	)	PUNCT
ajird-398	112	26	is	be	AUX
ajird-398	112	27	an	an	DET
ajird-398	112	28	analytic	analytic	ADJ
ajird-398	112	29	function	function	NOUN
ajird-398	112	30	.	.	PUNCT
ajird-398	113	1	solution	solution	NOUN
ajird-398	113	2	.	.	PUNCT
ajird-398	114	1	since	since	SCONJ
ajird-398	114	2	𝛼	𝛼	X
ajird-398	114	3	>	>	X
ajird-398	114	4	−1	−1	NOUN
ajird-398	114	5	and	and	CCONJ
ajird-398	114	6	𝑓(𝑥	𝑓(𝑥	NOUN
ajird-398	114	7	)	)	PUNCT
ajird-398	114	8	is	be	AUX
ajird-398	114	9	an	an	DET
ajird-398	114	10	analytic	analytic	ADJ
ajird-398	114	11	function	function	NOUN
ajird-398	114	12	in	in	ADP
ajird-398	114	13	the	the	DET
ajird-398	114	14	given	give	VERB
ajird-398	114	15	integral	integral	ADJ
ajird-398	114	16	,	,	PUNCT
ajird-398	114	17	the	the	DET
ajird-398	114	18	integral	integral	ADJ
ajird-398	114	19	depending	depend	VERB
ajird-398	114	20	on	on	ADP
ajird-398	114	21	this	this	DET
ajird-398	114	22	parameter	parameter	NOUN
ajird-398	114	23	is	be	AUX
ajird-398	114	24	convergent	convergent	ADJ
ajird-398	114	25	(	(	PUNCT
ajird-398	114	26	abel	abel	PROPN
ajird-398	114	27	's	's	PART
ajird-398	114	28	sign	sign	NOUN
ajird-398	114	29	)	)	PUNCT
ajird-398	114	30	.	.	PUNCT
ajird-398	115	1	also	also	ADV
ajird-398	115	2	,	,	PUNCT
ajird-398	115	3	the	the	DET
ajird-398	115	4	function	function	NOUN
ajird-398	115	5	𝑓(𝑥	𝑓(𝑥	NOUN
ajird-398	115	6	)	)	PUNCT
ajird-398	115	7	can	can	AUX
ajird-398	115	8	be	be	AUX
ajird-398	115	9	expanded	expand	VERB
ajird-398	115	10	into	into	ADP
ajird-398	115	11	a	a	DET
ajird-398	115	12	power	power	NOUN
ajird-398	115	13	series	series	NOUN
ajird-398	115	14	.	.	PUNCT
ajird-398	116	1	thus	thus	ADV
ajird-398	116	2	𝑓(𝑥	𝑓(𝑥	NOUN
ajird-398	116	3	)	)	PUNCT
ajird-398	116	4	=	=	SYM
ajird-398	116	5	∑	∑	PROPN
ajird-398	116	6	𝑎𝑛𝑥𝑛	𝑎𝑛𝑥𝑛	X
ajird-398	116	7	𝑛	𝑛	PROPN
ajird-398	116	8	!	!	NOUN
ajird-398	116	9	∞	∞	NUM
ajird-398	116	10	𝑛=1	𝑛=1	NOUN
ajird-398	116	11	.	.	PUNCT
ajird-398	117	1	here	here	ADV
ajird-398	117	2	𝑎𝑛	𝑎𝑛	VERB
ajird-398	117	3	−	−	PROPN
ajird-398	117	4	coefficients	coefficient	NOUN
ajird-398	117	5	of	of	ADP
ajird-398	117	6	extension	extension	NOUN
ajird-398	117	7	.	.	PUNCT
ajird-398	118	1	then	then	ADV
ajird-398	118	2	american	american	PROPN
ajird-398	118	3	journal	journal	PROPN
ajird-398	118	4	of	of	ADP
ajird-398	118	5	interdisciplinary	interdisciplinary	ADJ
ajird-398	118	6	research	research	NOUN
ajird-398	118	7	and	and	CCONJ
ajird-398	118	8	development	development	NOUN
ajird-398	118	9	issn	issn	PROPN
ajird-398	118	10	online	online	NOUN
ajird-398	118	11	:	:	PUNCT
ajird-398	118	12	2771	2771	NUM
ajird-398	118	13	-	-	SYM
ajird-398	118	14	8948	8948	NUM
ajird-398	118	15	website	website	NOUN
ajird-398	118	16	:	:	PUNCT
ajird-398	118	17	www.ajird.journalspark.org	www.ajird.journalspark.org	ADJ
ajird-398	118	18	volume	volume	NOUN
ajird-398	118	19	11	11	NUM
ajird-398	118	20	,	,	PUNCT
ajird-398	118	21	dec	dec	PROPN
ajird-398	118	22	.	.	PROPN
ajird-398	118	23	,	,	PUNCT
ajird-398	118	24	2022	2022	NUM
ajird-398	118	25	39	39	NUM
ajird-398	119	1	|	|	ADV
ajird-398	119	2	p	p	NOUN
ajird-398	119	3	a	a	DET
ajird-398	119	4	g	g	NOUN
ajird-398	119	5	e	e	NOUN
ajird-398	119	6	∫𝑓(𝑥)(𝑏	∫𝑓(𝑥)(𝑏	NOUN
ajird-398	119	7	−	−	PROPN
ajird-398	119	8	𝑥)𝛼	𝑥)𝛼	PUNCT
ajird-398	119	9	𝑏	𝑏	NOUN
ajird-398	119	10	0	0	NUM
ajird-398	119	11	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	119	12	=	=	SYM
ajird-398	119	13	∫	∫	PROPN
ajird-398	119	14	∑	∑	PROPN
ajird-398	119	15	𝑎𝑛𝑥𝑛	𝑎𝑛𝑥𝑛	PROPN
ajird-398	119	16	𝑛	𝑛	PROPN
ajird-398	119	17	!	!	NOUN
ajird-398	119	18	∞	∞	NUM
ajird-398	119	19	𝑛=1	𝑛=1	NOUN
ajird-398	119	20	(	(	PUNCT
ajird-398	119	21	𝑏	𝑏	NOUN
ajird-398	119	22	−	−	PROPN
ajird-398	119	23	𝑥)𝛼	𝑥)𝛼	PUNCT
ajird-398	119	24	𝑏	𝑏	NOUN
ajird-398	119	25	0	0	NUM
ajird-398	119	26	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	119	27	=	=	SYM
ajird-398	119	28	∑	∑	PUNCT
ajird-398	119	29	𝑎𝑛	𝑎𝑛	PROPN
ajird-398	119	30	𝑛	𝑛	PROPN
ajird-398	119	31	!	!	PUNCT
ajird-398	120	1	∫𝑥𝑛(𝑏	∫𝑥𝑛(𝑏	NUM
ajird-398	120	2	−	−	ADP
ajird-398	120	3	𝑥)𝛼	𝑥)𝛼	ADJ
ajird-398	120	4	𝑏	𝑏	NOUN
ajird-398	120	5	0	0	NUM
ajird-398	120	6	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	120	7	∞	∞	NUM
ajird-398	120	8	𝑛=0	𝑛=0	PROPN
ajird-398	120	9	=	=	PUNCT
ajird-398	121	1	=	=	PUNCT
ajird-398	121	2	[	[	PUNCT
ajird-398	121	3	𝑥	𝑥	X
ajird-398	121	4	=	=	PUNCT
ajird-398	121	5	𝑏𝑡	𝑏𝑡	NOUN
ajird-398	121	6	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	121	7	=	=	SYM
ajird-398	121	8	𝑏𝑑𝑡	𝑏𝑑𝑡	NOUN
ajird-398	121	9	𝑥	𝑥	NOUN
ajird-398	121	10	=	=	SYM
ajird-398	121	11	𝑏	𝑏	PROPN
ajird-398	121	12	→	→	SYM
ajird-398	121	13	𝑡	𝑡	X
ajird-398	121	14	=	=	SYM
ajird-398	121	15	1	1	NUM
ajird-398	121	16	𝑥	𝑥	NOUN
ajird-398	121	17	=	=	SYM
ajird-398	121	18	0	0	PUNCT
ajird-398	121	19	→	→	SYM
ajird-398	121	20	𝑡	𝑡	X
ajird-398	121	21	=	=	NOUN
ajird-398	121	22	0	0	PUNCT
ajird-398	121	23	]	]	PUNCT
ajird-398	122	1	=	=	PUNCT
ajird-398	122	2	∑	∑	PUNCT
ajird-398	122	3	𝑎𝑛	𝑎𝑛	PROPN
ajird-398	122	4	𝑛	𝑛	PROPN
ajird-398	122	5	!	!	PUNCT
ajird-398	123	1	∫𝑏𝑛𝑡𝑛(𝑏	∫𝑏𝑛𝑡𝑛(𝑏	NOUN
ajird-398	124	1	−	−	NUM
ajird-398	124	2	𝑏𝑡)𝛼	𝑏𝑡)𝛼	PROPN
ajird-398	124	3	1	1	NUM
ajird-398	124	4	0	0	NUM
ajird-398	124	5	𝑏𝑑𝑡	𝑏𝑑𝑡	PROPN
ajird-398	124	6	∞	∞	PROPN
ajird-398	124	7	𝑛=1	𝑛=1	NOUN
ajird-398	124	8	=	=	PUNCT
ajird-398	124	9	∑	∑	PUNCT
ajird-398	124	10	𝑎𝑛𝑏𝑛+𝛼+1	𝑎𝑛𝑏𝑛+𝛼+1	PROPN
ajird-398	124	11	𝑛	𝑛	PROPN
ajird-398	124	12	!	!	PROPN
ajird-398	124	13	∫	∫	PROPN
ajird-398	125	1	𝑡𝑛(1	𝑡𝑛(1	PROPN
ajird-398	125	2	−	−	PROPN
ajird-398	125	3	𝑡)𝛼	𝑡)𝛼	NOUN
ajird-398	125	4	1	1	NUM
ajird-398	125	5	0	0	NUM
ajird-398	125	6	𝑑𝑡	𝑑𝑡	ADP
ajird-398	125	7	∞	∞	PROPN
ajird-398	125	8	𝑛=1	𝑛=1	NOUN
ajird-398	125	9	=	=	NOUN
ajird-398	125	10	=	=	PUNCT
ajird-398	125	11	∑	∑	PUNCT
ajird-398	125	12	𝑎𝑛𝑏𝑛+𝛼+1	𝑎𝑛𝑏𝑛+𝛼+1	PROPN
ajird-398	125	13	𝑛	𝑛	PROPN
ajird-398	125	14	!	!	PUNCT
ajird-398	126	1	𝐵(𝑛	𝐵(𝑛	PUNCT
ajird-398	127	1	+	+	NUM
ajird-398	127	2	1	1	NUM
ajird-398	127	3	;	;	PUNCT
ajird-398	127	4	𝛼	𝛼	X
ajird-398	127	5	+	+	NOUN
ajird-398	127	6	1	1	NUM
ajird-398	127	7	)	)	PUNCT
ajird-398	127	8	∞	∞	NUM
ajird-398	127	9	𝑛=1	𝑛=1	NOUN
ajird-398	127	10	=	=	PUNCT
ajird-398	127	11	∑	∑	PUNCT
ajird-398	127	12	𝑎𝑛𝑏𝑛+𝛼+1	𝑎𝑛𝑏𝑛+𝛼+1	NOUN
ajird-398	127	13	𝑛	𝑛	PROPN
ajird-398	127	14	!	!	PUNCT
ajird-398	128	1	∙	∙	PROPN
ajird-398	128	2	г(𝑛	г(𝑛	PROPN
ajird-398	129	1	+	+	CCONJ
ajird-398	129	2	1	1	X
ajird-398	129	3	)	)	PUNCT
ajird-398	129	4	∙	∙	PROPN
ajird-398	129	5	г(𝛼	г(𝛼	PROPN
ajird-398	130	1	+	+	CCONJ
ajird-398	130	2	1	1	X
ajird-398	130	3	)	)	PUNCT
ajird-398	130	4	г(𝑛	г(𝑛	NOUN
ajird-398	131	1	+	+	CCONJ
ajird-398	131	2	𝛼	𝛼	PRON
ajird-398	131	3	+	+	ADJ
ajird-398	131	4	2	2	NUM
ajird-398	131	5	)	)	PUNCT
ajird-398	131	6	∞	∞	NUM
ajird-398	131	7	𝑛=1	𝑛=1	NOUN
ajird-398	132	1	=	=	NOUN
ajird-398	132	2	=	=	PUNCT
ajird-398	132	3	∑	∑	PUNCT
ajird-398	132	4	𝑎𝑛𝑏𝑛+𝛼+1	𝑎𝑛𝑏𝑛+𝛼+1	ADP
ajird-398	132	5	∙	∙	PROPN
ajird-398	132	6	𝛼г(𝛼	𝛼г(𝛼	NUM
ajird-398	132	7	)	)	PUNCT
ajird-398	132	8	(	(	PUNCT
ajird-398	132	9	𝑛	𝑛	PROPN
ajird-398	132	10	+	+	NUM
ajird-398	132	11	𝛼	𝛼	PROPN
ajird-398	132	12	+	+	NUM
ajird-398	132	13	2)(𝑛	2)(𝑛	NUM
ajird-398	132	14	+	+	SYM
ajird-398	132	15	𝛼)г(𝑛	𝛼)г(𝑛	PROPN
ajird-398	132	16	+	+	NUM
ajird-398	132	17	𝛼	𝛼	X
ajird-398	132	18	)	)	PUNCT
ajird-398	132	19	∞	∞	NUM
ajird-398	132	20	𝑛=1	𝑛=1	NOUN
ajird-398	132	21	.	.	PUNCT
ajird-398	133	1	problem	problem	NOUN
ajird-398	133	2	4	4	NUM
ajird-398	133	3	.	.	PUNCT
ajird-398	133	4	evaluate	evaluate	VERB
ajird-398	133	5	the	the	DET
ajird-398	133	6	value	value	NOUN
ajird-398	133	7	of	of	ADP
ajird-398	133	8	the	the	DET
ajird-398	133	9	following	follow	VERB
ajird-398	133	10	integral	integral	ADJ
ajird-398	133	11	:	:	PUNCT
ajird-398	133	12	𝐼(𝛼	𝐼(𝛼	NOUN
ajird-398	133	13	)	)	PUNCT
ajird-398	133	14	=	=	SYM
ajird-398	134	1	∫	∫	PROPN
ajird-398	134	2	√1	√1	ADV
ajird-398	135	1	−	−	PROPN
ajird-398	135	2	𝑥	𝑥	PRON
ajird-398	135	3	sin	sin	NOUN
ajird-398	135	4	𝑎𝑥	𝑎𝑥	NOUN
ajird-398	135	5	1	1	NUM
ajird-398	135	6	0	0	NUM
ajird-398	135	7	𝑑𝑥.	𝑑𝑥.	NOUN
ajird-398	135	8	solution	solution	NOUN
ajird-398	135	9	.	.	PUNCT
ajird-398	136	1	to	to	PART
ajird-398	136	2	calculate	calculate	VERB
ajird-398	136	3	this	this	DET
ajird-398	136	4	integral	integral	ADJ
ajird-398	136	5	,	,	PUNCT
ajird-398	136	6	we	we	PRON
ajird-398	136	7	use	use	VERB
ajird-398	136	8	the	the	DET
ajird-398	136	9	following	follow	VERB
ajird-398	136	10	taylor	taylor	PROPN
ajird-398	136	11	expansion	expansion	NOUN
ajird-398	136	12	:	:	PUNCT
ajird-398	136	13	sin	sin	NOUN
ajird-398	136	14	𝛼𝑥	𝛼𝑥	ADV
ajird-398	136	15	=	=	SYM
ajird-398	136	16	∑	∑	PROPN
ajird-398	136	17	(	(	PUNCT
ajird-398	136	18	−1)𝑛𝛼2𝑛+1	−1)𝑛𝛼2𝑛+1	PROPN
ajird-398	136	19	𝑥2𝑛+1	𝑥2𝑛+1	PROPN
ajird-398	136	20	(	(	PUNCT
ajird-398	136	21	2𝑛+1	2𝑛+1	NOUN
ajird-398	136	22	)	)	PUNCT
ajird-398	136	23	!	!	PUNCT
ajird-398	137	1	∞	∞	NUM
ajird-398	137	2	𝑛=1	𝑛=1	NOUN
ajird-398	137	3	.	.	PUNCT
ajird-398	138	1	then	then	ADV
ajird-398	138	2	,	,	PUNCT
ajird-398	138	3	𝐼(𝛼	𝐼(𝛼	NOUN
ajird-398	138	4	)	)	PUNCT
ajird-398	138	5	=	=	SYM
ajird-398	138	6	∫√1	∫√1	PROPN
ajird-398	138	7	−	−	NOUN
ajird-398	138	8	𝑥	𝑥	PRON
ajird-398	138	9	∙	∙	PROPN
ajird-398	138	10	sin	sin	NOUN
ajird-398	138	11	𝑎𝑥	𝑎𝑥	ADP
ajird-398	138	12	1	1	NUM
ajird-398	138	13	0	0	NUM
ajird-398	138	14	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	138	15	=	=	SYM
ajird-398	138	16	∫	∫	PROPN
ajird-398	138	17	∑	∑	PROPN
ajird-398	138	18	(	(	PUNCT
ajird-398	138	19	−1)𝑛𝛼2𝑛+1𝑥2𝑛+1	−1)𝑛𝛼2𝑛+1𝑥2𝑛+1	PROPN
ajird-398	138	20	(	(	PUNCT
ajird-398	138	21	2𝑛	2𝑛	PROPN
ajird-398	138	22	+	+	PROPN
ajird-398	138	23	1	1	NUM
ajird-398	138	24	)	)	PUNCT
ajird-398	138	25	!	!	PUNCT
ajird-398	139	1	∞	∞	NUM
ajird-398	140	1	𝑛=0	𝑛=0	PROPN
ajird-398	141	1	√1	√1	ADV
ajird-398	141	2	−	−	PROPN
ajird-398	141	3	𝑥	𝑥	NOUN
ajird-398	141	4	1	1	NUM
ajird-398	141	5	0	0	NUM
ajird-398	141	6	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	141	7	=	=	SYM
ajird-398	141	8	=	=	PUNCT
ajird-398	141	9	∑	∑	PUNCT
ajird-398	141	10	(	(	PUNCT
ajird-398	141	11	−1)𝑛𝛼2𝑛+1	−1)𝑛𝛼2𝑛+1	PROPN
ajird-398	141	12	(	(	PUNCT
ajird-398	141	13	2𝑛	2𝑛	PROPN
ajird-398	141	14	+	+	PROPN
ajird-398	141	15	1	1	NUM
ajird-398	141	16	)	)	PUNCT
ajird-398	141	17	!	!	PUNCT
ajird-398	142	1	∫𝑥2𝑛+1(1	∫𝑥2𝑛+1(1	X
ajird-398	142	2	−	−	PRON
ajird-398	142	3	𝑥	𝑥	NOUN
ajird-398	142	4	)	)	PUNCT
ajird-398	142	5	1	1	NUM
ajird-398	142	6	2	2	NUM
ajird-398	142	7	1	1	NUM
ajird-398	142	8	0	0	NUM
ajird-398	142	9	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	142	10	∞	∞	NUM
ajird-398	142	11	𝑛=0	𝑛=0	PROPN
ajird-398	142	12	=	=	PUNCT
ajird-398	142	13	∑	∑	PUNCT
ajird-398	142	14	(	(	PUNCT
ajird-398	142	15	−1)𝑛𝛼2𝑛+1	−1)𝑛𝛼2𝑛+1	PROPN
ajird-398	142	16	(	(	PUNCT
ajird-398	142	17	2𝑛	2𝑛	PROPN
ajird-398	142	18	+	+	PROPN
ajird-398	142	19	1	1	NUM
ajird-398	142	20	)	)	PUNCT
ajird-398	142	21	!	!	PUNCT
ajird-398	143	1	∙	∙	PROPN
ajird-398	143	2	𝐵	𝐵	PROPN
ajird-398	143	3	(	(	PUNCT
ajird-398	143	4	2𝑛	2𝑛	PROPN
ajird-398	143	5	+	+	CCONJ
ajird-398	143	6	2	2	NUM
ajird-398	143	7	,	,	PUNCT
ajird-398	143	8	3	3	NUM
ajird-398	143	9	2	2	NUM
ajird-398	143	10	)	)	PUNCT
ajird-398	143	11	∞	∞	NUM
ajird-398	143	12	𝑛=0	𝑛=0	PROPN
ajird-398	143	13	=	=	PUNCT
ajird-398	143	14	∑	∑	PUNCT
ajird-398	143	15	(	(	PUNCT
ajird-398	143	16	−1)𝑛𝛼2𝑛+1	−1)𝑛𝛼2𝑛+1	PROPN
ajird-398	143	17	(	(	PUNCT
ajird-398	143	18	2𝑛	2𝑛	PROPN
ajird-398	143	19	+	+	PROPN
ajird-398	143	20	1	1	NUM
ajird-398	143	21	)	)	PUNCT
ajird-398	143	22	!	!	PUNCT
ajird-398	144	1	∙	∙	NOUN
ajird-398	144	2	г(2𝑛	г(2𝑛	NUM
ajird-398	145	1	+	+	SYM
ajird-398	145	2	2	2	X
ajird-398	145	3	)	)	PUNCT
ajird-398	145	4	∙	∙	PROPN
ajird-398	145	5	г	г	PROPN
ajird-398	145	6	(	(	PUNCT
ajird-398	145	7	3	3	NUM
ajird-398	145	8	2	2	NUM
ajird-398	145	9	)	)	PUNCT
ajird-398	145	10	г	г	PROPN
ajird-398	145	11	(	(	PUNCT
ajird-398	145	12	2𝑛	2𝑛	NOUN
ajird-398	145	13	+	+	CCONJ
ajird-398	145	14	7	7	NUM
ajird-398	145	15	2	2	NUM
ajird-398	145	16	)	)	PUNCT
ajird-398	145	17	∞	∞	NUM
ajird-398	145	18	𝑛=0	𝑛=0	PROPN
ajird-398	145	19	=	=	PUNCT
ajird-398	145	20	∑	∑	PUNCT
ajird-398	145	21	(	(	PUNCT
ajird-398	145	22	−1)𝑛𝛼2𝑛+1	−1)𝑛𝛼2𝑛+1	PROPN
ajird-398	145	23	(	(	PUNCT
ajird-398	145	24	2𝑛	2𝑛	PROPN
ajird-398	145	25	+	+	PROPN
ajird-398	145	26	1	1	NUM
ajird-398	145	27	)	)	PUNCT
ajird-398	145	28	!	!	PUNCT
ajird-398	146	1	∙	∙	PROPN
ajird-398	146	2	(	(	PUNCT
ajird-398	146	3	2𝑛	2𝑛	NOUN
ajird-398	146	4	+	+	PROPN
ajird-398	146	5	1	1	NUM
ajird-398	146	6	)	)	PUNCT
ajird-398	146	7	!	!	PUNCT
ajird-398	147	1	∙	∙	NOUN
ajird-398	147	2	1	1	NUM
ajird-398	147	3	2	2	NUM
ajird-398	147	4	(	(	PUNCT
ajird-398	147	5	(	(	PUNCT
ajird-398	147	6	2	2	NUM
ajird-398	147	7	∙	∙	PROPN
ajird-398	147	8	(	(	PUNCT
ajird-398	147	9	2𝑛	2𝑛	NOUN
ajird-398	147	10	+	+	CCONJ
ajird-398	147	11	2))‼√𝜋	2))‼√𝜋	NUM
ajird-398	147	12	)	)	PUNCT
ajird-398	147	13	22𝑛+2	22𝑛+2	NUM
ajird-398	147	14	∞	∞	NUM
ajird-398	147	15	𝑛=0	𝑛=0	PROPN
ajird-398	148	1	=	=	PUNCT
ajird-398	148	2	∑	∑	PUNCT
ajird-398	148	3	(	(	PUNCT
ajird-398	148	4	−1)𝑛	−1)𝑛	X
ajird-398	148	5	∙	∙	PROPN
ajird-398	148	6	𝛼2𝑛+1	𝛼2𝑛+1	NOUN
ajird-398	148	7	∙	∙	PROPN
ajird-398	148	8	22𝑛+2	22𝑛+2	PROPN
ajird-398	148	9	22𝑛+1	22𝑛+1	NUM
ajird-398	148	10	∙	∙	PROPN
ajird-398	148	11	𝑛	𝑛	PROPN
ajird-398	148	12	!	!	PROPN
ajird-398	148	13	√𝜋	√𝜋	X
ajird-398	149	1	∞	∞	NUM
ajird-398	149	2	𝑛=0	𝑛=0	X
ajird-398	149	3	=	=	SYM
ajird-398	149	4	2	2	NUM
ajird-398	149	5	√𝜋	√𝜋	NUM
ajird-398	149	6	∑	∑	PUNCT
ajird-398	149	7	(	(	PUNCT
ajird-398	149	8	−1)𝑛	−1)𝑛	X
ajird-398	149	9	∙	∙	PROPN
ajird-398	149	10	𝛼2𝑛+1	𝛼2𝑛+1	PROPN
ajird-398	149	11	𝑛	𝑛	PROPN
ajird-398	149	12	!	!	NOUN
ajird-398	149	13	∞	∞	NUM
ajird-398	149	14	𝑛=0	𝑛=0	PROPN
ajird-398	149	15	.	.	PUNCT
ajird-398	150	1	problem	problem	NOUN
ajird-398	150	2	5	5	NUM
ajird-398	150	3	.	.	PUNCT
ajird-398	150	4	evaluate	evaluate	VERB
ajird-398	150	5	the	the	DET
ajird-398	150	6	value	value	NOUN
ajird-398	150	7	of	of	ADP
ajird-398	150	8	the	the	DET
ajird-398	150	9	following	follow	VERB
ajird-398	150	10	integral	integral	ADJ
ajird-398	150	11	:	:	PUNCT
ajird-398	150	12	𝐼(𝛼	𝐼(𝛼	NOUN
ajird-398	150	13	)	)	PUNCT
ajird-398	150	14	=	=	SYM
ajird-398	150	15	∫	∫	PROPN
ajird-398	150	16	𝑥𝛼𝑥1	𝑥𝛼𝑥1	PROPN
ajird-398	150	17	0	0	NUM
ajird-398	151	1	𝑑𝑥.	𝑑𝑥.	NOUN
ajird-398	151	2	solution	solution	NOUN
ajird-398	151	3	.	.	PUNCT
ajird-398	152	1	to	to	PART
ajird-398	152	2	calculate	calculate	VERB
ajird-398	152	3	this	this	DET
ajird-398	152	4	integral	integral	ADJ
ajird-398	152	5	,	,	PUNCT
ajird-398	152	6	we	we	PRON
ajird-398	152	7	first	first	ADV
ajird-398	152	8	get	get	VERB
ajird-398	152	9	the	the	DET
ajird-398	152	10	following	follow	VERB
ajird-398	152	11	substitution	substitution	NOUN
ajird-398	152	12	:	:	PUNCT
ajird-398	152	13	𝐼(𝛼	𝐼(𝛼	NOUN
ajird-398	152	14	)	)	PUNCT
ajird-398	152	15	=	=	SYM
ajird-398	152	16	∫𝑥𝛼𝑥	∫𝑥𝛼𝑥	NUM
ajird-398	152	17	1	1	NUM
ajird-398	152	18	0	0	NUM
ajird-398	152	19	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	152	20	=	=	SYM
ajird-398	152	21	[	[	PUNCT
ajird-398	152	22	𝑥	𝑥	X
ajird-398	152	23	=	=	PUNCT
ajird-398	152	24	𝑒−𝜉	𝑒−𝜉	NOUN
ajird-398	152	25	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	152	26	=	=	PUNCT
ajird-398	152	27	−𝑒−𝜉𝑑𝜉	−𝑒−𝜉𝑑𝜉	NOUN
ajird-398	152	28	𝑥	𝑥	PROPN
ajird-398	152	29	→	→	SYM
ajird-398	152	30	0	0	NUM
ajird-398	152	31	𝜉	𝜉	NOUN
ajird-398	152	32	→	→	SYM
ajird-398	152	33	∞	∞	NUM
ajird-398	152	34	𝑥	𝑥	X
ajird-398	152	35	→	→	SYM
ajird-398	152	36	1	1	NUM
ajird-398	152	37	𝜉	𝜉	NOUN
ajird-398	152	38	→	→	SYM
ajird-398	152	39	0	0	NUM
ajird-398	152	40	]	]	PUNCT
ajird-398	153	1	=	=	SYM
ajird-398	153	2	∫	∫	PROPN
ajird-398	153	3	𝑒−𝜉	𝑒−𝜉	VERB
ajird-398	153	4	∙	∙	PROPN
ajird-398	153	5	𝑒−𝜉𝛼𝑒−𝜉	𝑒−𝜉𝛼𝑒−𝜉	ADP
ajird-398	153	6	+	+	PROPN
ajird-398	153	7	∞	∞	NOUN
ajird-398	153	8	0	0	NUM
ajird-398	154	1	𝑑𝜉.	𝑑𝜉.	NOUN
ajird-398	154	2	now	now	ADV
ajird-398	154	3	we	we	PRON
ajird-398	154	4	use	use	VERB
ajird-398	154	5	the	the	DET
ajird-398	154	6	following	follow	VERB
ajird-398	154	7	taylor	taylor	PROPN
ajird-398	154	8	expansion	expansion	NOUN
ajird-398	154	9	:	:	PUNCT
ajird-398	154	10	𝑒−𝑥	𝑒−𝑥	NOUN
ajird-398	154	11	=	=	PUNCT
ajird-398	154	12	∑	∑	PUNCT
ajird-398	154	13	(	(	PUNCT
ajird-398	154	14	−1)𝑛𝑥𝑛	−1)𝑛𝑥𝑛	NOUN
ajird-398	154	15	𝑛	𝑛	PROPN
ajird-398	154	16	!	!	NOUN
ajird-398	154	17	∞	∞	NUM
ajird-398	154	18	𝑛=0	𝑛=0	PROPN
ajird-398	154	19	.	.	PUNCT
ajird-398	155	1	then	then	ADV
ajird-398	155	2	,	,	PUNCT
ajird-398	155	3	american	american	PROPN
ajird-398	155	4	journal	journal	PROPN
ajird-398	155	5	of	of	ADP
ajird-398	155	6	interdisciplinary	interdisciplinary	ADJ
ajird-398	155	7	research	research	NOUN
ajird-398	155	8	and	and	CCONJ
ajird-398	155	9	development	development	NOUN
ajird-398	155	10	issn	issn	PROPN
ajird-398	155	11	online	online	NOUN
ajird-398	155	12	:	:	PUNCT
ajird-398	155	13	2771	2771	NUM
ajird-398	155	14	-	-	SYM
ajird-398	155	15	8948	8948	NUM
ajird-398	155	16	website	website	NOUN
ajird-398	155	17	:	:	PUNCT
ajird-398	155	18	www.ajird.journalspark.org	www.ajird.journalspark.org	ADJ
ajird-398	155	19	volume	volume	NOUN
ajird-398	155	20	11	11	NUM
ajird-398	155	21	,	,	PUNCT
ajird-398	155	22	dec	dec	PROPN
ajird-398	155	23	.	.	PROPN
ajird-398	155	24	,	,	PUNCT
ajird-398	155	25	2022	2022	NUM
ajird-398	155	26	40	40	NUM
ajird-398	156	1	|	|	ADV
ajird-398	156	2	p	p	X
ajird-398	156	3	a	a	DET
ajird-398	156	4	g	g	NOUN
ajird-398	156	5	e	e	X
ajird-398	156	6	𝐼(𝛼	𝐼(𝛼	NOUN
ajird-398	156	7	)	)	PUNCT
ajird-398	156	8	=	=	SYM
ajird-398	156	9	∫	∫	PROPN
ajird-398	156	10	𝑒−𝜉	𝑒−𝜉	PROPN
ajird-398	156	11	∑	∑	X
ajird-398	156	12	(	(	PUNCT
ajird-398	156	13	−1)𝑛(𝛼𝜉𝑒−𝜉	−1)𝑛(𝛼𝜉𝑒−𝜉	NOUN
ajird-398	156	14	)	)	PUNCT
ajird-398	156	15	𝑛	𝑛	PRON
ajird-398	156	16	𝑛	𝑛	NOUN
ajird-398	156	17	!	!	PUNCT
ajird-398	156	18	∞	∞	NUM
ajird-398	157	1	𝑛=0	𝑛=0	X
ajird-398	158	1	+	+	ADJ
ajird-398	158	2	∞	∞	NOUN
ajird-398	158	3	0	0	NUM
ajird-398	159	1	𝑑𝜉	𝑑𝜉	ADP
ajird-398	159	2	=	=	PRON
ajird-398	159	3	∑	∑	PROPN
ajird-398	159	4	(	(	PUNCT
ajird-398	159	5	−1)𝑛𝛼𝑛	−1)𝑛𝛼𝑛	PROPN
ajird-398	159	6	𝑛	𝑛	PROPN
ajird-398	159	7	!	!	PROPN
ajird-398	159	8	∫	∫	PROPN
ajird-398	159	9	𝜉𝑛𝑒−𝜉(𝑛+1	𝜉𝑛𝑒−𝜉(𝑛+1	PROPN
ajird-398	159	10	)	)	PUNCT
ajird-398	160	1	+	+	NOUN
ajird-398	160	2	∞	∞	NOUN
ajird-398	160	3	0	0	NUM
ajird-398	160	4	𝑑𝜉	𝑑𝜉	ADP
ajird-398	160	5	∞	∞	NUM
ajird-398	160	6	𝑛=0	𝑛=0	PROPN
ajird-398	161	1	=	=	PUNCT
ajird-398	161	2	=	=	PUNCT
ajird-398	162	1	[	[	PUNCT
ajird-398	162	2	𝜉(𝑛	𝜉(𝑛	NOUN
ajird-398	162	3	+	+	CCONJ
ajird-398	162	4	1	1	X
ajird-398	162	5	)	)	PUNCT
ajird-398	162	6	=	=	SYM
ajird-398	162	7	𝜓	𝜓	X
ajird-398	162	8	𝑑𝜉	𝑑𝜉	ADP
ajird-398	162	9	=	=	PUNCT
ajird-398	162	10	𝑑𝜓	𝑑𝜓	PROPN
ajird-398	162	11	𝑛	𝑛	PROPN
ajird-398	162	12	+	+	NOUN
ajird-398	162	13	1	1	NUM
ajird-398	162	14	]	]	PUNCT
ajird-398	162	15	=	=	PUNCT
ajird-398	162	16	∑	∑	PUNCT
ajird-398	162	17	(	(	PUNCT
ajird-398	162	18	−1)𝑛𝛼𝑛	−1)𝑛𝛼𝑛	PROPN
ajird-398	162	19	𝑛	𝑛	PROPN
ajird-398	162	20	!	!	PROPN
ajird-398	162	21	∫	∫	PROPN
ajird-398	162	22	(	(	PUNCT
ajird-398	162	23	𝜓	𝜓	PROPN
ajird-398	162	24	𝑛	𝑛	PROPN
ajird-398	162	25	+	+	NOUN
ajird-398	162	26	1	1	NUM
ajird-398	162	27	)	)	PUNCT
ajird-398	162	28	𝑛	𝑛	NOUN
ajird-398	162	29	𝑒−𝜓	𝑒−𝜓	NOUN
ajird-398	162	30	𝑛	𝑛	PRON
ajird-398	162	31	+	+	NOUN
ajird-398	162	32	1	1	NUM
ajird-398	162	33	+	+	NUM
ajird-398	162	34	∞	∞	NUM
ajird-398	162	35	0	0	NUM
ajird-398	162	36	𝑑𝜓	𝑑𝜓	NUM
ajird-398	162	37	∞	∞	NUM
ajird-398	162	38	𝑛=0	𝑛=0	PROPN
ajird-398	162	39	=	=	PUNCT
ajird-398	163	1	=	=	PUNCT
ajird-398	163	2	∑	∑	PUNCT
ajird-398	163	3	(	(	PUNCT
ajird-398	163	4	−1)𝑛𝛼𝑛	−1)𝑛𝛼𝑛	PROPN
ajird-398	163	5	𝑛	𝑛	PROPN
ajird-398	163	6	!	!	PUNCT
ajird-398	163	7	(	(	PUNCT
ajird-398	163	8	𝑛	𝑛	PROPN
ajird-398	163	9	+	+	SYM
ajird-398	163	10	1)𝑛+1	1)𝑛+1	NUM
ajird-398	163	11	∫	∫	NOUN
ajird-398	163	12	𝜓𝑛𝑒−𝜓	𝜓𝑛𝑒−𝜓	PROPN
ajird-398	164	1	+	+	NOUN
ajird-398	164	2	∞	∞	PROPN
ajird-398	164	3	0	0	NUM
ajird-398	165	1	𝑑𝜓	𝑑𝜓	PROPN
ajird-398	166	1	+	+	NOUN
ajird-398	166	2	∞	∞	NOUN
ajird-398	166	3	𝑛=0	𝑛=0	NOUN
ajird-398	166	4	=	=	PUNCT
ajird-398	166	5	∑	∑	PUNCT
ajird-398	166	6	(	(	PUNCT
ajird-398	166	7	−1)𝑛𝛼𝑛	−1)𝑛𝛼𝑛	PROPN
ajird-398	166	8	𝑛	𝑛	PROPN
ajird-398	166	9	!	!	PUNCT
ajird-398	166	10	(	(	PUNCT
ajird-398	166	11	𝑛	𝑛	PROPN
ajird-398	166	12	+	+	NUM
ajird-398	166	13	1)𝑛+1	1)𝑛+1	NUM
ajird-398	166	14	г(𝑛	г(𝑛	NOUN
ajird-398	166	15	+	+	CCONJ
ajird-398	166	16	1	1	X
ajird-398	166	17	)	)	PUNCT
ajird-398	166	18	∞	∞	NUM
ajird-398	166	19	𝑛=0	𝑛=0	PROPN
ajird-398	166	20	=	=	PUNCT
ajird-398	166	21	∑	∑	PUNCT
ajird-398	166	22	(	(	PUNCT
ajird-398	166	23	−1)𝑛−1𝛼𝑛	−1)𝑛−1𝛼𝑛	PROPN
ajird-398	166	24	𝑛𝑛	𝑛𝑛	NOUN
ajird-398	166	25	∞	∞	PROPN
ajird-398	166	26	𝑛=1	𝑛=1	NOUN
ajird-398	166	27	.	.	PUNCT
ajird-398	167	1	thus	thus	ADV
ajird-398	167	2	,	,	PUNCT
ajird-398	167	3	following	follow	VERB
ajird-398	167	4	equality	equality	NOUN
ajird-398	167	5	is	be	AUX
ajird-398	167	6	holds	hold	NOUN
ajird-398	167	7	for	for	ADP
ajird-398	167	8	given	give	VERB
ajird-398	167	9	integral	integral	ADJ
ajird-398	167	10	:	:	PUNCT
ajird-398	167	11	𝐼(𝛼	𝐼(𝛼	NOUN
ajird-398	167	12	)	)	PUNCT
ajird-398	167	13	=	=	PUNCT
ajird-398	167	14	∑	∑	PUNCT
ajird-398	167	15	(	(	PUNCT
ajird-398	167	16	−1)𝑛−1𝛼𝑛	−1)𝑛−1𝛼𝑛	PROPN
ajird-398	167	17	𝑛𝑛	𝑛𝑛	NOUN
ajird-398	167	18	∞	∞	PROPN
ajird-398	167	19	𝑛=1	𝑛=1	NOUN
ajird-398	167	20	.	.	PUNCT
ajird-398	168	1	in	in	ADP
ajird-398	168	2	particular	particular	ADJ
ajird-398	168	3	,	,	PUNCT
ajird-398	168	4	∫	∫	PROPN
ajird-398	168	5	𝑥𝑥1	𝑥𝑥1	NOUN
ajird-398	168	6	0	0	PUNCT
ajird-398	168	7	𝑑𝑥	𝑑𝑥	VERB
ajird-398	168	8	=	=	SYM
ajird-398	168	9	∑	∑	PUNCT
ajird-398	168	10	(	(	PUNCT
ajird-398	168	11	−1)𝑛−1	−1)𝑛−1	NOUN
ajird-398	168	12	𝑛𝑛	𝑛𝑛	NOUN
ajird-398	168	13	∞	∞	PROPN
ajird-398	168	14	𝑛=1	𝑛=1	NOUN
ajird-398	168	15	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
ajird-398	168	16	𝛼	𝛼	NOUN
ajird-398	168	17	=	=	SYM
ajird-398	168	18	1	1	NUM
ajird-398	168	19	,	,	PUNCT
ajird-398	168	20	and	and	CCONJ
ajird-398	168	21	∫	∫	PROPN
ajird-398	168	22	1	1	NUM
ajird-398	168	23	𝑥𝑥	𝑥𝑥	ADP
ajird-398	168	24	1	1	NUM
ajird-398	168	25	0	0	NUM
ajird-398	168	26	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	168	27	=	=	SYM
ajird-398	168	28	∑	∑	PROPN
ajird-398	168	29	1	1	NUM
ajird-398	168	30	𝑛𝑛	𝑛𝑛	NUM
ajird-398	168	31	∞	∞	PROPN
ajird-398	168	32	𝑛=1	𝑛=1	NOUN
ajird-398	168	33	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
ajird-398	168	34	𝛼	𝛼	NOUN
ajird-398	168	35	=	=	NOUN
ajird-398	168	36	1	1	NUM
ajird-398	168	37	.	.	PUNCT
ajird-398	168	38	problem	problem	NOUN
ajird-398	168	39	6	6	NUM
ajird-398	168	40	.	.	PUNCT
ajird-398	168	41	evaluate	evaluate	VERB
ajird-398	168	42	the	the	DET
ajird-398	168	43	value	value	NOUN
ajird-398	168	44	of	of	ADP
ajird-398	168	45	the	the	DET
ajird-398	168	46	following	follow	VERB
ajird-398	168	47	integral	integral	ADJ
ajird-398	168	48	:	:	PUNCT
ajird-398	168	49	𝜓(𝛼	𝜓(𝛼	PROPN
ajird-398	168	50	)	)	PUNCT
ajird-398	169	1	=	=	SYM
ajird-398	169	2	∫	∫	PROPN
ajird-398	169	3	𝑒−𝛼𝑥2+𝑥	𝑒−𝛼𝑥2+𝑥	PROPN
ajird-398	169	4	sin	sin	NOUN
ajird-398	169	5	𝑥	𝑥	PROPN
ajird-398	170	1	+	+	NOUN
ajird-398	170	2	∞	∞	NOUN
ajird-398	170	3	0	0	NUM
ajird-398	170	4	.	.	PUNCT
ajird-398	171	1	solution	solution	NOUN
ajird-398	171	2	.	.	PUNCT
ajird-398	172	1	it	it	PRON
ajird-398	172	2	is	be	AUX
ajird-398	172	3	known	know	VERB
ajird-398	172	4	that	that	SCONJ
ajird-398	172	5	the	the	DET
ajird-398	172	6	given	give	VERB
ajird-398	172	7	integral	integral	ADJ
ajird-398	172	8	depending	depending	NOUN
ajird-398	172	9	on	on	ADP
ajird-398	172	10	the	the	DET
ajird-398	172	11	parameter	parameter	NOUN
ajird-398	172	12	is	be	AUX
ajird-398	172	13	uniformly	uniformly	ADV
ajird-398	172	14	convergent	convergent	NOUN
ajird-398	172	15	for	for	ADP
ajird-398	172	16	𝑎	𝑎	PROPN
ajird-398	172	17	>	>	X
ajird-398	172	18	0	0	PUNCT
ajird-398	173	1	according	accord	VERB
ajird-398	173	2	to	to	ADP
ajird-398	173	3	the	the	DET
ajird-398	173	4	comparison	comparison	NOUN
ajird-398	173	5	property	property	NOUN
ajird-398	173	6	.	.	PUNCT
ajird-398	174	1	so	so	ADV
ajird-398	174	2	,	,	PUNCT
ajird-398	174	3	the	the	DET
ajird-398	174	4	sign	sign	NOUN
ajird-398	174	5	of	of	ADP
ajird-398	174	6	integral	integral	ADJ
ajird-398	174	7	can	can	AUX
ajird-398	174	8	be	be	AUX
ajird-398	174	9	replaced	replace	VERB
ajird-398	174	10	by	by	ADP
ajird-398	174	11	the	the	DET
ajird-398	174	12	sign	sign	NOUN
ajird-398	174	13	of	of	ADP
ajird-398	174	14	sum	sum	NOUN
ajird-398	174	15	.	.	PUNCT
ajird-398	175	1	to	to	PART
ajird-398	175	2	calculate	calculate	VERB
ajird-398	175	3	the	the	DET
ajird-398	175	4	value	value	NOUN
ajird-398	175	5	of	of	ADP
ajird-398	175	6	this	this	DET
ajird-398	175	7	integral	integral	ADJ
ajird-398	175	8	,	,	PUNCT
ajird-398	175	9	we	we	PRON
ajird-398	175	10	use	use	VERB
ajird-398	175	11	the	the	DET
ajird-398	175	12	following	follow	VERB
ajird-398	175	13	taylor	taylor	PROPN
ajird-398	175	14	expansion	expansion	NOUN
ajird-398	175	15	:	:	PUNCT
ajird-398	175	16	𝑒𝑥	𝑒𝑥	NOUN
ajird-398	175	17	sin	sin	NOUN
ajird-398	175	18	𝑥	𝑥	PROPN
ajird-398	175	19	=	=	SYM
ajird-398	175	20	∑	∑	PROPN
ajird-398	175	21	2	2	NUM
ajird-398	175	22	𝑛	𝑛	DET
ajird-398	175	23	2	2	NUM
ajird-398	175	24	sin	sin	NOUN
ajird-398	175	25	𝑛𝜋	𝑛𝜋	ADP
ajird-398	175	26	4	4	NUM
ajird-398	175	27	𝑛	𝑛	NOUN
ajird-398	175	28	!	!	PUNCT
ajird-398	176	1	∙	∙	X
ajird-398	176	2	𝑥𝑛∞	𝑥𝑛∞	PROPN
ajird-398	176	3	𝑛=0	𝑛=0	PROPN
ajird-398	176	4	.	.	PUNCT
ajird-398	177	1	in	in	ADP
ajird-398	177	2	that	that	DET
ajird-398	177	3	case	case	NOUN
ajird-398	177	4	,	,	PUNCT
ajird-398	177	5	𝜓(𝛼	𝜓(𝛼	PROPN
ajird-398	177	6	)	)	PUNCT
ajird-398	178	1	=	=	SYM
ajird-398	178	2	∫	∫	PROPN
ajird-398	179	1	𝑒−𝛼𝑥2	𝑒−𝛼𝑥2	INTJ
ajird-398	179	2	∑	∑	ADP
ajird-398	179	3	2	2	NUM
ajird-398	179	4	𝑛	𝑛	DET
ajird-398	179	5	2	2	NUM
ajird-398	179	6	sin	sin	NOUN
ajird-398	179	7	𝑛𝜋	𝑛𝜋	ADP
ajird-398	179	8	4	4	NUM
ajird-398	179	9	𝑛	𝑛	NOUN
ajird-398	179	10	!	!	PUNCT
ajird-398	180	1	∙	∙	NOUN
ajird-398	180	2	𝑥𝑛	𝑥𝑛	PROPN
ajird-398	180	3	∞	∞	NUM
ajird-398	180	4	𝑛=0	𝑛=0	X
ajird-398	181	1	+	+	NOUN
ajird-398	181	2	∞	∞	NOUN
ajird-398	181	3	0	0	NUM
ajird-398	181	4	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	181	5	=	=	SYM
ajird-398	181	6	∑	∑	ADP
ajird-398	181	7	2	2	NUM
ajird-398	181	8	𝑛	𝑛	DET
ajird-398	181	9	2	2	NUM
ajird-398	181	10	sin	sin	NOUN
ajird-398	181	11	𝑛𝜋	𝑛𝜋	ADP
ajird-398	181	12	4	4	NUM
ajird-398	181	13	𝑛	𝑛	NOUN
ajird-398	181	14	!	!	PUNCT
ajird-398	181	15	∫	∫	PROPN
ajird-398	182	1	𝑒−𝛼𝑥2	𝑒−𝛼𝑥2	PROPN
ajird-398	183	1	∙	∙	PROPN
ajird-398	183	2	𝑥𝑛	𝑥𝑛	PROPN
ajird-398	184	1	+	+	ADJ
ajird-398	184	2	∞	∞	NOUN
ajird-398	184	3	0	0	NUM
ajird-398	184	4	𝑑𝑥	𝑑𝑥	VERB
ajird-398	184	5	+	+	NOUN
ajird-398	184	6	∞	∞	NOUN
ajird-398	184	7	𝑛=0	𝑛=0	NOUN
ajird-398	185	1	=	=	PUNCT
ajird-398	186	1	=	=	PUNCT
ajird-398	187	1	[	[	PUNCT
ajird-398	187	2	𝛼𝑥2	𝛼𝑥2	NUM
ajird-398	187	3	=	=	SYM
ajird-398	187	4	𝜉	𝜉	NOUN
ajird-398	187	5	𝑥	𝑥	NOUN
ajird-398	187	6	=	=	NOUN
ajird-398	187	7	√	√	NUM
ajird-398	187	8	𝜉	𝜉	VERB
ajird-398	187	9	𝛼	𝛼	NOUN
ajird-398	187	10	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	187	11	=	=	SYM
ajird-398	187	12	1	1	NUM
ajird-398	187	13	2√𝜉𝛼	2√𝜉𝛼	NUM
ajird-398	187	14	𝑑𝜉	𝑑𝜉	ADP
ajird-398	187	15	]	]	PUNCT
ajird-398	187	16	=	=	PUNCT
ajird-398	187	17	∑	∑	PROPN
ajird-398	187	18	2	2	NUM
ajird-398	187	19	𝑛	𝑛	DET
ajird-398	187	20	2	2	NUM
ajird-398	187	21	sin	sin	NOUN
ajird-398	187	22	𝑛𝜋	𝑛𝜋	ADP
ajird-398	187	23	4	4	NUM
ajird-398	187	24	𝑛	𝑛	NOUN
ajird-398	187	25	!	!	PUNCT
ajird-398	187	26	∫	∫	PROPN
ajird-398	187	27	𝑒−𝜉	𝑒−𝜉	PROPN
ajird-398	187	28	(	(	PUNCT
ajird-398	187	29	𝜉	𝜉	PROPN
ajird-398	187	30	𝛼	𝛼	NOUN
ajird-398	187	31	)	)	PUNCT
ajird-398	187	32	𝑛	𝑛	PRON
ajird-398	187	33	2	2	NUM
ajird-398	187	34	+	+	NOUN
ajird-398	187	35	∞	∞	NOUN
ajird-398	187	36	0	0	NUM
ajird-398	188	1	∙	∙	NOUN
ajird-398	188	2	1	1	NUM
ajird-398	188	3	2√𝜉𝛼	2√𝜉𝛼	NUM
ajird-398	188	4	𝑑𝜉	𝑑𝜉	ADP
ajird-398	188	5	+	+	ADJ
ajird-398	188	6	∞	∞	NOUN
ajird-398	188	7	𝑛=0	𝑛=0	NOUN
ajird-398	188	8	=	=	PUNCT
ajird-398	188	9	=	=	SYM
ajird-398	188	10	1	1	NUM
ajird-398	188	11	2	2	NUM
ajird-398	188	12	∑	∑	SYM
ajird-398	188	13	2	2	NUM
ajird-398	188	14	𝑛	𝑛	DET
ajird-398	188	15	2	2	NUM
ajird-398	188	16	sin	sin	NOUN
ajird-398	188	17	𝑛𝜋	𝑛𝜋	ADP
ajird-398	188	18	4	4	NUM
ajird-398	188	19	𝑛	𝑛	NOUN
ajird-398	188	20	!	!	PUNCT
ajird-398	188	21	𝛼	𝛼	PROPN
ajird-398	188	22	𝑛+1	𝑛+1	PROPN
ajird-398	188	23	2	2	NUM
ajird-398	188	24	∫	∫	NOUN
ajird-398	188	25	𝑒−𝜉	𝑒−𝜉	X
ajird-398	188	26	+	+	ADV
ajird-398	188	27	∞	∞	NOUN
ajird-398	188	28	0	0	NUM
ajird-398	189	1	∙	∙	PROPN
ajird-398	189	2	𝜉	𝜉	X
ajird-398	189	3	𝑛−1	𝑛−1	NUM
ajird-398	189	4	2	2	NUM
ajird-398	189	5	𝑑𝜉	𝑑𝜉	ADP
ajird-398	189	6	+	+	PROPN
ajird-398	189	7	∞	∞	PROPN
ajird-398	189	8	𝑛=0	𝑛=0	NOUN
ajird-398	189	9	=	=	SYM
ajird-398	189	10	1	1	NUM
ajird-398	189	11	2	2	NUM
ajird-398	189	12	∑	∑	SYM
ajird-398	189	13	2	2	NUM
ajird-398	189	14	𝑛	𝑛	DET
ajird-398	189	15	2	2	NUM
ajird-398	189	16	sin	sin	NOUN
ajird-398	189	17	𝑛𝜋	𝑛𝜋	ADP
ajird-398	189	18	4	4	NUM
ajird-398	189	19	𝑛	𝑛	NOUN
ajird-398	189	20	!	!	PUNCT
ajird-398	189	21	𝛼	𝛼	PROPN
ajird-398	189	22	𝑛+1	𝑛+1	PROPN
ajird-398	189	23	2	2	NUM
ajird-398	189	24	г	г	PROPN
ajird-398	189	25	(	(	PUNCT
ajird-398	189	26	𝑛	𝑛	PROPN
ajird-398	189	27	+	+	NOUN
ajird-398	189	28	1	1	NUM
ajird-398	189	29	2	2	NUM
ajird-398	189	30	)	)	PUNCT
ajird-398	190	1	+	+	NOUN
ajird-398	190	2	∞	∞	NUM
ajird-398	190	3	𝑛=0	𝑛=0	PROPN
ajird-398	190	4	.	.	PUNCT
ajird-398	191	1	probem	probem	NOUN
ajird-398	191	2	7	7	NUM
ajird-398	191	3	.	.	PUNCT
ajird-398	191	4	evaluate	evaluate	VERB
ajird-398	191	5	the	the	DET
ajird-398	191	6	value	value	NOUN
ajird-398	191	7	of	of	ADP
ajird-398	191	8	the	the	DET
ajird-398	191	9	following	follow	VERB
ajird-398	191	10	integral	integral	ADJ
ajird-398	191	11	:	:	PUNCT
ajird-398	191	12	𝐼(𝛽	𝐼(𝛽	NOUN
ajird-398	191	13	)	)	PUNCT
ajird-398	191	14	=	=	SYM
ajird-398	192	1	∫	∫	PROPN
ajird-398	192	2	𝑒−𝛼𝑥	𝑒−𝛼𝑥	NOUN
ajird-398	192	3	𝛽+𝑥	𝛽+𝑥	PROPN
ajird-398	193	1	+	+	PROPN
ajird-398	193	2	∞	∞	PROPN
ajird-398	193	3	0	0	NUM
ajird-398	193	4	𝑑𝑥.	𝑑𝑥.	NOUN
ajird-398	193	5	solution	solution	NOUN
ajird-398	193	6	.	.	PUNCT
ajird-398	194	1	it	it	PRON
ajird-398	194	2	is	be	AUX
ajird-398	194	3	known	know	VERB
ajird-398	194	4	that	that	SCONJ
ajird-398	194	5	the	the	DET
ajird-398	194	6	given	give	VERB
ajird-398	194	7	integral	integral	ADJ
ajird-398	194	8	depending	depending	NOUN
ajird-398	194	9	on	on	ADP
ajird-398	194	10	the	the	DET
ajird-398	194	11	parameter	parameter	NOUN
ajird-398	194	12	is	be	AUX
ajird-398	194	13	uniformly	uniformly	ADV
ajird-398	194	14	convergent	convergent	NOUN
ajird-398	194	15	according	accord	VERB
ajird-398	194	16	to	to	ADP
ajird-398	194	17	the	the	DET
ajird-398	194	18	comparison	comparison	NOUN
ajird-398	194	19	property	property	NOUN
ajird-398	194	20	.	.	PUNCT
ajird-398	195	1	so	so	ADV
ajird-398	195	2	,	,	PUNCT
ajird-398	195	3	the	the	DET
ajird-398	195	4	sign	sign	NOUN
ajird-398	195	5	of	of	ADP
ajird-398	195	6	integral	integral	ADJ
ajird-398	195	7	can	can	AUX
ajird-398	195	8	be	be	AUX
ajird-398	195	9	replaced	replace	VERB
ajird-398	195	10	by	by	ADP
ajird-398	195	11	american	american	ADJ
ajird-398	195	12	journal	journal	PROPN
ajird-398	195	13	of	of	ADP
ajird-398	195	14	interdisciplinary	interdisciplinary	ADJ
ajird-398	195	15	research	research	NOUN
ajird-398	195	16	and	and	CCONJ
ajird-398	195	17	development	development	NOUN
ajird-398	195	18	issn	issn	PROPN
ajird-398	195	19	online	online	NOUN
ajird-398	195	20	:	:	PUNCT
ajird-398	195	21	2771	2771	NUM
ajird-398	195	22	-	-	SYM
ajird-398	195	23	8948	8948	NUM
ajird-398	195	24	website	website	NOUN
ajird-398	195	25	:	:	PUNCT
ajird-398	195	26	www.ajird.journalspark.org	www.ajird.journalspark.org	ADJ
ajird-398	195	27	volume	volume	NOUN
ajird-398	195	28	11	11	NUM
ajird-398	195	29	,	,	PUNCT
ajird-398	195	30	dec	dec	PROPN
ajird-398	195	31	.	.	PROPN
ajird-398	195	32	,	,	PUNCT
ajird-398	195	33	2022	2022	NUM
ajird-398	195	34	41	41	NUM
ajird-398	196	1	|	|	ADV
ajird-398	196	2	p	p	ADP
ajird-398	196	3	a	a	DET
ajird-398	196	4	g	g	NOUN
ajird-398	196	5	e	e	NOUN
ajird-398	196	6	the	the	DET
ajird-398	196	7	sign	sign	NOUN
ajird-398	196	8	of	of	ADP
ajird-398	196	9	sum	sum	NOUN
ajird-398	196	10	.	.	PUNCT
ajird-398	197	1	to	to	PART
ajird-398	197	2	calculate	calculate	VERB
ajird-398	197	3	the	the	DET
ajird-398	197	4	value	value	NOUN
ajird-398	197	5	of	of	ADP
ajird-398	197	6	this	this	DET
ajird-398	197	7	integral	integral	ADJ
ajird-398	197	8	,	,	PUNCT
ajird-398	197	9	we	we	PRON
ajird-398	197	10	use	use	VERB
ajird-398	197	11	the	the	DET
ajird-398	197	12	following	follow	VERB
ajird-398	197	13	taylor	taylor	PROPN
ajird-398	197	14	expansion	expansion	NOUN
ajird-398	197	15	:	:	PUNCT
ajird-398	197	16	1	1	NUM
ajird-398	197	17	1+𝛽𝑥	1+𝛽𝑥	NUM
ajird-398	197	18	=	=	SYM
ajird-398	197	19	∑	∑	PROPN
ajird-398	197	20	(	(	PUNCT
ajird-398	197	21	−1)𝑛𝛽𝑛𝑥𝑛∞	−1)𝑛𝛽𝑛𝑥𝑛∞	PROPN
ajird-398	197	22	𝑛=0	𝑛=0	PROPN
ajird-398	197	23	.	.	PUNCT
ajird-398	198	1	in	in	ADP
ajird-398	198	2	that	that	DET
ajird-398	198	3	case	case	NOUN
ajird-398	198	4	,	,	PUNCT
ajird-398	198	5	𝐼(𝛽	𝐼(𝛽	NOUN
ajird-398	198	6	)	)	PUNCT
ajird-398	198	7	=	=	SYM
ajird-398	198	8	∫	∫	PROPN
ajird-398	198	9	𝑒−𝛼𝑥	𝑒−𝛼𝑥	PROPN
ajird-398	198	10	∙	∙	PROPN
ajird-398	198	11	∑(−1)𝑛𝛽𝑛𝑥𝑛	∑(−1)𝑛𝛽𝑛𝑥𝑛	NOUN
ajird-398	198	12	∞	∞	PROPN
ajird-398	198	13	𝑛=0	𝑛=0	X
ajird-398	198	14	+	+	NOUN
ajird-398	198	15	∞	∞	NOUN
ajird-398	198	16	0	0	NUM
ajird-398	198	17	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	198	18	=	=	SYM
ajird-398	198	19	∑(−1)𝑛𝛽𝑛	∑(−1)𝑛𝛽𝑛	PROPN
ajird-398	198	20	∫	∫	PROPN
ajird-398	198	21	𝑥𝑛𝑒−𝛼𝑥	𝑥𝑛𝑒−𝛼𝑥	X
ajird-398	199	1	+	+	PROPN
ajird-398	199	2	∞	∞	NOUN
ajird-398	199	3	0	0	NUM
ajird-398	199	4	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	199	5	∞	∞	NUM
ajird-398	199	6	𝑛=0	𝑛=0	PROPN
ajird-398	199	7	=	=	PUNCT
ajird-398	200	1	[	[	PUNCT
ajird-398	200	2	𝛼𝑥	𝛼𝑥	ADV
ajird-398	200	3	=	=	NOUN
ajird-398	200	4	𝜙	𝜙	NOUN
ajird-398	200	5	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	200	6	=	=	PRON
ajird-398	200	7	𝑑𝜙	𝑑𝜙	ADP
ajird-398	200	8	𝛼	𝛼	NOUN
ajird-398	200	9	]	]	X
ajird-398	200	10	=	=	PUNCT
ajird-398	201	1	=	=	SYM
ajird-398	201	2	∑	∑	PUNCT
ajird-398	201	3	(	(	PUNCT
ajird-398	201	4	−1)𝑛𝛽𝑛	−1)𝑛𝛽𝑛	PROPN
ajird-398	201	5	𝛼𝑛+1	𝛼𝑛+1	NUM
ajird-398	201	6	∫	∫	NOUN
ajird-398	201	7	𝜙𝑛𝑒−𝜙	𝜙𝑛𝑒−𝜙	VERB
ajird-398	202	1	+	+	NOUN
ajird-398	202	2	∞	∞	PROPN
ajird-398	202	3	0	0	NUM
ajird-398	202	4	∞	∞	NUM
ajird-398	202	5	𝑛=0	𝑛=0	NOUN
ajird-398	202	6	𝑑𝜙	𝑑𝜙	ADP
ajird-398	202	7	=	=	PUNCT
ajird-398	202	8	∑	∑	PROPN
ajird-398	202	9	(	(	PUNCT
ajird-398	202	10	−1)𝑛𝛽𝑛	−1)𝑛𝛽𝑛	PROPN
ajird-398	202	11	𝛼𝑛+1	𝛼𝑛+1	NUM
ajird-398	202	12	г(𝑛	г(𝑛	NOUN
ajird-398	202	13	+	+	CCONJ
ajird-398	202	14	1	1	X
ajird-398	202	15	)	)	PUNCT
ajird-398	202	16	∞	∞	NUM
ajird-398	203	1	𝑛=0	𝑛=0	PROPN
ajird-398	203	2	=	=	PUNCT
ajird-398	203	3	∑	∑	PUNCT
ajird-398	203	4	(	(	PUNCT
ajird-398	203	5	−1)𝑛𝛽𝑛𝑛	−1)𝑛𝛽𝑛𝑛	PROPN
ajird-398	203	6	!	!	PUNCT
ajird-398	204	1	𝛼𝑛+1	𝛼𝑛+1	NUM
ajird-398	204	2	∞	∞	NUM
ajird-398	204	3	𝑛=0	𝑛=0	PROPN
ajird-398	204	4	.	.	PUNCT
ajird-398	205	1	thus	thus	ADV
ajird-398	205	2	,	,	PUNCT
ajird-398	205	3	𝐼(𝛽	𝐼(𝛽	NOUN
ajird-398	205	4	)	)	PUNCT
ajird-398	205	5	=	=	SYM
ajird-398	205	6	∑	∑	PUNCT
ajird-398	205	7	(	(	PUNCT
ajird-398	205	8	−1)𝑛𝛽𝑛𝑛	−1)𝑛𝛽𝑛𝑛	PROPN
ajird-398	205	9	!	!	PUNCT
ajird-398	206	1	𝛼𝑛+1	𝛼𝑛+1	NUM
ajird-398	206	2	∞	∞	NUM
ajird-398	206	3	𝑛=0	𝑛=0	PROPN
ajird-398	206	4	.	.	PUNCT
ajird-398	207	1	problem	problem	NOUN
ajird-398	207	2	8	8	NUM
ajird-398	207	3	.	.	PUNCT
ajird-398	207	4	evaluate	evaluate	VERB
ajird-398	207	5	the	the	DET
ajird-398	207	6	value	value	NOUN
ajird-398	207	7	of	of	ADP
ajird-398	207	8	the	the	DET
ajird-398	207	9	following	follow	VERB
ajird-398	207	10	integral	integral	ADJ
ajird-398	207	11	:	:	PUNCT
ajird-398	207	12	∫	∫	PROPN
ajird-398	207	13	𝑒−𝛼𝑥𝑝	𝑒−𝛼𝑥𝑝	VERB
ajird-398	207	14	sin	sin	PROPN
ajird-398	207	15	𝛽𝑥𝑞+∞	𝛽𝑥𝑞+∞	PROPN
ajird-398	207	16	0	0	NUM
ajird-398	207	17	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	207	18	,	,	PUNCT
ajird-398	207	19	here	here	ADV
ajird-398	207	20	𝑝	𝑝	ADP
ajird-398	207	21	>	>	ADP
ajird-398	207	22	0	0	NUM
ajird-398	207	23	,	,	PUNCT
ajird-398	207	24	𝑞	𝑞	X
ajird-398	207	25	>	>	X
ajird-398	207	26	−1	−1	NOUN
ajird-398	207	27	.	.	PUNCT
ajird-398	208	1	solution	solution	NOUN
ajird-398	208	2	.	.	PUNCT
ajird-398	209	1	it	it	PRON
ajird-398	209	2	is	be	AUX
ajird-398	209	3	known	know	VERB
ajird-398	209	4	that	that	SCONJ
ajird-398	209	5	the	the	DET
ajird-398	209	6	given	give	VERB
ajird-398	209	7	integral	integral	ADJ
ajird-398	209	8	depending	depending	NOUN
ajird-398	209	9	on	on	ADP
ajird-398	209	10	the	the	DET
ajird-398	209	11	parameter	parameter	NOUN
ajird-398	209	12	is	be	AUX
ajird-398	209	13	uniformly	uniformly	ADV
ajird-398	209	14	convergent	convergent	NOUN
ajird-398	209	15	for	for	ADP
ajird-398	209	16	𝑎	𝑎	PROPN
ajird-398	209	17	>	>	X
ajird-398	209	18	0	0	PUNCT
ajird-398	210	1	according	accord	VERB
ajird-398	210	2	to	to	ADP
ajird-398	210	3	the	the	DET
ajird-398	210	4	comparison	comparison	NOUN
ajird-398	210	5	property	property	NOUN
ajird-398	210	6	.	.	PUNCT
ajird-398	211	1	so	so	ADV
ajird-398	211	2	,	,	PUNCT
ajird-398	211	3	the	the	DET
ajird-398	211	4	sign	sign	NOUN
ajird-398	211	5	of	of	ADP
ajird-398	211	6	integral	integral	ADJ
ajird-398	211	7	can	can	AUX
ajird-398	211	8	be	be	AUX
ajird-398	211	9	replaced	replace	VERB
ajird-398	211	10	by	by	ADP
ajird-398	211	11	the	the	DET
ajird-398	211	12	sign	sign	NOUN
ajird-398	211	13	of	of	ADP
ajird-398	211	14	sum	sum	NOUN
ajird-398	211	15	.	.	PUNCT
ajird-398	212	1	to	to	PART
ajird-398	212	2	calculate	calculate	VERB
ajird-398	212	3	the	the	DET
ajird-398	212	4	value	value	NOUN
ajird-398	212	5	of	of	ADP
ajird-398	212	6	this	this	DET
ajird-398	212	7	integral	integral	ADJ
ajird-398	212	8	,	,	PUNCT
ajird-398	212	9	we	we	PRON
ajird-398	212	10	use	use	VERB
ajird-398	212	11	the	the	DET
ajird-398	212	12	following	follow	VERB
ajird-398	212	13	taylor	taylor	PROPN
ajird-398	212	14	expansion	expansion	NOUN
ajird-398	212	15	:	:	PUNCT
ajird-398	212	16	sin	sin	NOUN
ajird-398	212	17	𝑥	𝑥	NOUN
ajird-398	212	18	=	=	PUNCT
ajird-398	212	19	∑	∑	PROPN
ajird-398	212	20	(	(	PUNCT
ajird-398	212	21	−1)𝑛−1𝑥2𝑛−1	−1)𝑛−1𝑥2𝑛−1	X
ajird-398	212	22	(	(	PUNCT
ajird-398	212	23	2𝑛−1	2𝑛−1	NOUN
ajird-398	212	24	)	)	PUNCT
ajird-398	212	25	!	!	PUNCT
ajird-398	213	1	∞	∞	NUM
ajird-398	213	2	𝑛=1	𝑛=1	NOUN
ajird-398	213	3	.	.	PUNCT
ajird-398	214	1	in	in	ADP
ajird-398	214	2	that	that	DET
ajird-398	214	3	case	case	NOUN
ajird-398	214	4	,	,	PUNCT
ajird-398	214	5	∫	∫	PROPN
ajird-398	214	6	𝑒−𝛼𝑥𝑝	𝑒−𝛼𝑥𝑝	PROPN
ajird-398	214	7	sin	sin	PROPN
ajird-398	214	8	𝛽𝑥𝑞	𝛽𝑥𝑞	PROPN
ajird-398	214	9	+	+	PROPN
ajird-398	214	10	∞	∞	NOUN
ajird-398	214	11	0	0	NUM
ajird-398	214	12	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	214	13	=	=	SYM
ajird-398	214	14	∫	∫	NOUN
ajird-398	214	15	𝑒−𝛼𝑥𝑝	𝑒−𝛼𝑥𝑝	NOUN
ajird-398	214	16	∑	∑	PROPN
ajird-398	214	17	(	(	PUNCT
ajird-398	214	18	−1)𝑛−1𝛽2𝑛−1𝑥𝑞(2𝑛−1	−1)𝑛−1𝛽2𝑛−1𝑥𝑞(2𝑛−1	NOUN
ajird-398	214	19	)	)	PUNCT
ajird-398	214	20	(	(	PUNCT
ajird-398	214	21	2𝑛	2𝑛	NOUN
ajird-398	214	22	−	−	PROPN
ajird-398	214	23	1	1	NUM
ajird-398	214	24	)	)	PUNCT
ajird-398	214	25	!	!	PUNCT
ajird-398	215	1	∞	∞	NUM
ajird-398	215	2	𝑛=1	𝑛=1	PRON
ajird-398	216	1	+	+	ADJ
ajird-398	216	2	∞	∞	NOUN
ajird-398	216	3	0	0	NUM
ajird-398	216	4	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	216	5	=	=	SYM
ajird-398	216	6	=	=	PUNCT
ajird-398	216	7	∑	∑	PUNCT
ajird-398	216	8	(	(	PUNCT
ajird-398	216	9	−1)𝑛−1𝛽2𝑛−1	−1)𝑛−1𝛽2𝑛−1	PROPN
ajird-398	216	10	(	(	PUNCT
ajird-398	216	11	2𝑛	2𝑛	NOUN
ajird-398	216	12	−	−	PROPN
ajird-398	216	13	1	1	NUM
ajird-398	216	14	)	)	PUNCT
ajird-398	216	15	!	!	PUNCT
ajird-398	217	1	∫	∫	PROPN
ajird-398	217	2	𝑒−𝛼𝑥𝑝	𝑒−𝛼𝑥𝑝	PROPN
ajird-398	217	3	𝑥𝑞(2𝑛−1	𝑥𝑞(2𝑛−1	PROPN
ajird-398	217	4	)	)	PUNCT
ajird-398	218	1	+	+	NOUN
ajird-398	218	2	∞	∞	NOUN
ajird-398	218	3	0	0	NUM
ajird-398	218	4	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	218	5	∞	∞	NUM
ajird-398	218	6	𝑛=1	𝑛=1	NOUN
ajird-398	218	7	=	=	SYM
ajird-398	218	8	[	[	PUNCT
ajird-398	218	9	𝛼𝑥𝑝	𝛼𝑥𝑝	NOUN
ajird-398	218	10	=	=	PUNCT
ajird-398	218	11	𝜉	𝜉	NOUN
ajird-398	218	12	𝑥	𝑥	NOUN
ajird-398	218	13	=	=	NOUN
ajird-398	218	14	√	√	NUM
ajird-398	218	15	𝜉	𝜉	NOUN
ajird-398	218	16	𝛼	𝛼	VERB
ajird-398	218	17	𝑝	𝑝	NOUN
ajird-398	218	18	𝑑𝑥	𝑑𝑥	VERB
ajird-398	218	19	=	=	SYM
ajird-398	218	20	𝜉	𝜉	ADP
ajird-398	218	21	1	1	NUM
ajird-398	218	22	𝑝	𝑝	NOUN
ajird-398	218	23	−1	−1	NOUN
ajird-398	218	24	∙	∙	PROPN
ajird-398	218	25	1	1	NUM
ajird-398	218	26	𝑝	𝑝	ADP
ajird-398	218	27	∙	∙	PROPN
ajird-398	218	28	𝛼	𝛼	PROPN
ajird-398	218	29	1	1	NUM
ajird-398	218	30	𝑝	𝑝	NOUN
ajird-398	218	31	𝑑𝜉	𝑑𝜉	ADP
ajird-398	218	32	]	]	PUNCT
ajird-398	218	33	=	=	PUNCT
ajird-398	218	34	=	=	SYM
ajird-398	218	35	∑	∑	PROPN
ajird-398	218	36	(	(	PUNCT
ajird-398	218	37	−1)𝑛−1𝛽(2𝑛−1	−1)𝑛−1𝛽(2𝑛−1	NOUN
ajird-398	218	38	)	)	PUNCT
ajird-398	218	39	(	(	PUNCT
ajird-398	218	40	2𝑛	2𝑛	NOUN
ajird-398	218	41	−	−	PROPN
ajird-398	218	42	1	1	NUM
ajird-398	218	43	)	)	PUNCT
ajird-398	218	44	!	!	PUNCT
ajird-398	219	1	∙	∙	PROPN
ajird-398	219	2	𝑝𝛼	𝑝𝛼	VERB
ajird-398	219	3	𝑞(2𝑛−1)+1	𝑞(2𝑛−1)+1	PROPN
ajird-398	219	4	𝑝	𝑝	PROPN
ajird-398	219	5	∫	∫	PROPN
ajird-398	219	6	𝑒−𝜉𝜉	𝑒−𝜉𝜉	PROPN
ajird-398	219	7	𝑞(2𝑛−1)+1	𝑞(2𝑛−1)+1	PROPN
ajird-398	219	8	𝑝	𝑝	ADP
ajird-398	219	9	−1	−1	NOUN
ajird-398	219	10	+	+	NOUN
ajird-398	219	11	∞	∞	NOUN
ajird-398	219	12	0	0	NUM
ajird-398	219	13	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	219	14	∞	∞	NUM
ajird-398	219	15	𝑛=1	𝑛=1	NOUN
ajird-398	219	16	=	=	NOUN
ajird-398	219	17	=	=	SYM
ajird-398	219	18	∑	∑	PROPN
ajird-398	219	19	(	(	PUNCT
ajird-398	219	20	−1)𝑛−1𝛽(2𝑛−1	−1)𝑛−1𝛽(2𝑛−1	NOUN
ajird-398	219	21	)	)	PUNCT
ajird-398	219	22	(	(	PUNCT
ajird-398	219	23	2𝑛	2𝑛	NOUN
ajird-398	219	24	−	−	PROPN
ajird-398	219	25	1	1	NUM
ajird-398	219	26	)	)	PUNCT
ajird-398	219	27	!	!	PUNCT
ajird-398	220	1	𝑝𝛼	𝑝𝛼	PROPN
ajird-398	220	2	𝑞(2𝑛−1)+1	𝑞(2𝑛−1)+1	PROPN
ajird-398	220	3	𝑝	𝑝	PROPN
ajird-398	220	4	∞	∞	PROPN
ajird-398	220	5	𝑛=1	𝑛=1	NOUN
ajird-398	220	6	г	г	PROPN
ajird-398	220	7	(	(	PUNCT
ajird-398	220	8	𝑞(2𝑛	𝑞(2𝑛	NUM
ajird-398	220	9	−	−	NOUN
ajird-398	220	10	1	1	NUM
ajird-398	220	11	)	)	PUNCT
ajird-398	220	12	+	+	CCONJ
ajird-398	220	13	1	1	NUM
ajird-398	220	14	𝑝	𝑝	NOUN
ajird-398	220	15	)	)	PUNCT
ajird-398	220	16	.	.	PUNCT
ajird-398	221	1	in	in	ADP
ajird-398	221	2	this	this	DET
ajird-398	221	3	case	case	NOUN
ajird-398	221	4	,	,	PUNCT
ajird-398	221	5	if	if	SCONJ
ajird-398	221	6	𝛼	𝛼	X
ajird-398	221	7	=	=	SYM
ajird-398	221	8	1	1	NUM
ajird-398	221	9	,	,	PUNCT
ajird-398	221	10	𝛽	𝛽	NOUN
ajird-398	221	11	=	=	SYM
ajird-398	221	12	1	1	NUM
ajird-398	221	13	,	,	PUNCT
ajird-398	221	14	𝑝	𝑝	NOUN
ajird-398	221	15	=	=	SYM
ajird-398	221	16	2	2	NUM
ajird-398	221	17	,	,	PUNCT
ajird-398	221	18	𝑞	𝑞	X
ajird-398	221	19	=	=	NOUN
ajird-398	221	20	4	4	NUM
ajird-398	221	21	,	,	PUNCT
ajird-398	221	22	we	we	PRON
ajird-398	221	23	get	get	VERB
ajird-398	221	24	the	the	DET
ajird-398	221	25	following	follow	VERB
ajird-398	221	26	equation	equation	NOUN
ajird-398	221	27	:	:	PUNCT
ajird-398	221	28	∫	∫	PROPN
ajird-398	221	29	𝑒−𝑥2	𝑒−𝑥2	ADP
ajird-398	221	30	sin	sin	NOUN
ajird-398	221	31	𝑥4	𝑥4	PROPN
ajird-398	222	1	+	+	PROPN
ajird-398	222	2	∞	∞	PROPN
ajird-398	222	3	0	0	NUM
ajird-398	222	4	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	222	5	=	=	SYM
ajird-398	222	6	∑	∑	PUNCT
ajird-398	222	7	(	(	PUNCT
ajird-398	222	8	−1)𝑛−1𝛽2𝑛−1	−1)𝑛−1𝛽2𝑛−1	PROPN
ajird-398	222	9	(	(	PUNCT
ajird-398	222	10	2𝑛	2𝑛	NOUN
ajird-398	222	11	−	−	PROPN
ajird-398	222	12	1	1	NUM
ajird-398	222	13	)	)	PUNCT
ajird-398	222	14	!	!	PUNCT
ajird-398	223	1	2𝛼	2𝛼	PROPN
ajird-398	223	2	(	(	PUNCT
ajird-398	223	3	4(2𝑛−1)+1	4(2𝑛−1)+1	NUM
ajird-398	223	4	)	)	PUNCT
ajird-398	223	5	2	2	NUM
ajird-398	223	6	г	г	PROPN
ajird-398	223	7	(	(	PUNCT
ajird-398	223	8	4(2𝑛	4(2𝑛	NUM
ajird-398	223	9	−	−	NOUN
ajird-398	223	10	1	1	NUM
ajird-398	223	11	)	)	PUNCT
ajird-398	223	12	+	+	CCONJ
ajird-398	223	13	1	1	NUM
ajird-398	223	14	2	2	NUM
ajird-398	223	15	)	)	PUNCT
ajird-398	223	16	∞	∞	NUM
ajird-398	223	17	𝑛=1	𝑛=1	NOUN
ajird-398	223	18	=	=	SYM
ajird-398	223	19	american	american	PROPN
ajird-398	223	20	journal	journal	PROPN
ajird-398	223	21	of	of	ADP
ajird-398	223	22	interdisciplinary	interdisciplinary	ADJ
ajird-398	223	23	research	research	NOUN
ajird-398	223	24	and	and	CCONJ
ajird-398	223	25	development	development	NOUN
ajird-398	223	26	issn	issn	PROPN
ajird-398	223	27	online	online	NOUN
ajird-398	223	28	:	:	PUNCT
ajird-398	223	29	2771	2771	NUM
ajird-398	223	30	-	-	SYM
ajird-398	223	31	8948	8948	NUM
ajird-398	223	32	website	website	NOUN
ajird-398	223	33	:	:	PUNCT
ajird-398	223	34	www.ajird.journalspark.org	www.ajird.journalspark.org	ADJ
ajird-398	223	35	volume	volume	NOUN
ajird-398	223	36	11	11	NUM
ajird-398	223	37	,	,	PUNCT
ajird-398	223	38	dec	dec	PROPN
ajird-398	223	39	.	.	PROPN
ajird-398	223	40	,	,	PUNCT
ajird-398	223	41	2022	2022	NUM
ajird-398	223	42	42	42	NUM
ajird-398	224	1	|	|	ADV
ajird-398	224	2	p	p	ADP
ajird-398	224	3	a	a	DET
ajird-398	224	4	g	g	NOUN
ajird-398	224	5	e	e	NOUN
ajird-398	224	6	=	=	PUNCT
ajird-398	224	7	∑	∑	PUNCT
ajird-398	224	8	(	(	PUNCT
ajird-398	224	9	−1)𝑛−1𝛽2𝑛−1	−1)𝑛−1𝛽2𝑛−1	NOUN
ajird-398	224	10	2(2𝑛	2(2𝑛	NUM
ajird-398	224	11	−	−	NOUN
ajird-398	224	12	1	1	NUM
ajird-398	224	13	)	)	PUNCT
ajird-398	224	14	!	!	PUNCT
ajird-398	225	1	𝛼4𝑛−	𝛼4𝑛−	NOUN
ajird-398	225	2	3	3	NUM
ajird-398	225	3	2	2	NUM
ajird-398	225	4	г((4𝑛	г((4𝑛	NOUN
ajird-398	225	5	−	−	NOUN
ajird-398	225	6	2	2	NUM
ajird-398	225	7	)	)	PUNCT
ajird-398	225	8	+	+	CCONJ
ajird-398	225	9	1	1	NUM
ajird-398	225	10	2	2	NUM
ajird-398	225	11	)	)	PUNCT
ajird-398	225	12	∞	∞	NUM
ajird-398	225	13	𝑛=1	𝑛=1	NOUN
ajird-398	225	14	=	=	PUNCT
ajird-398	225	15	∑	∑	PUNCT
ajird-398	225	16	(	(	PUNCT
ajird-398	225	17	−1)𝑛−1𝛽2𝑛−1	−1)𝑛−1𝛽2𝑛−1	NOUN
ajird-398	225	18	2(2𝑛	2(2𝑛	NUM
ajird-398	225	19	−	−	NOUN
ajird-398	225	20	1	1	NUM
ajird-398	225	21	)	)	PUNCT
ajird-398	225	22	!	!	PUNCT
ajird-398	226	1	𝛼4𝑛−	𝛼4𝑛−	NOUN
ajird-398	226	2	3	3	NUM
ajird-398	226	3	2	2	NUM
ajird-398	226	4	∙	∙	X
ajird-398	226	5	(	(	PUNCT
ajird-398	226	6	2(4𝑛	2(4𝑛	NUM
ajird-398	226	7	−	−	NOUN
ajird-398	226	8	2	2	NUM
ajird-398	226	9	)	)	PUNCT
ajird-398	226	10	−	−	PROPN
ajird-398	226	11	1)‼	1)‼	PROPN
ajird-398	226	12	√𝜋	√𝜋	X
ajird-398	226	13	24𝑛−2	24𝑛−2	NUM
ajird-398	226	14	∞	∞	PROPN
ajird-398	226	15	𝑛=1	𝑛=1	NOUN
ajird-398	226	16	=	=	PUNCT
ajird-398	226	17	∑	∑	PUNCT
ajird-398	226	18	(	(	PUNCT
ajird-398	226	19	−1)𝑛−1𝛽2𝑛−1	−1)𝑛−1𝛽2𝑛−1	NOUN
ajird-398	226	20	2(2𝑛	2(2𝑛	NUM
ajird-398	226	21	−	−	NOUN
ajird-398	226	22	1	1	NUM
ajird-398	226	23	)	)	PUNCT
ajird-398	226	24	!	!	PUNCT
ajird-398	227	1	𝛼4𝑛−	𝛼4𝑛−	NOUN
ajird-398	227	2	3	3	NUM
ajird-398	227	3	2	2	NUM
ajird-398	227	4	∙	∙	X
ajird-398	227	5	(	(	PUNCT
ajird-398	227	6	8𝑛	8𝑛	NOUN
ajird-398	227	7	−	−	PROPN
ajird-398	227	8	3)‼√𝜋	3)‼√𝜋	NUM
ajird-398	227	9	2𝑛	2𝑛	PROPN
ajird-398	227	10	∞	∞	NUM
ajird-398	227	11	𝑛=1	𝑛=1	NOUN
ajird-398	227	12	.	.	PUNCT
ajird-398	228	1	problem	problem	NOUN
ajird-398	228	2	9	9	NUM
ajird-398	228	3	.	.	PUNCT
ajird-398	228	4	evaluate	evaluate	VERB
ajird-398	228	5	the	the	DET
ajird-398	228	6	value	value	NOUN
ajird-398	228	7	of	of	ADP
ajird-398	228	8	the	the	DET
ajird-398	228	9	following	follow	VERB
ajird-398	228	10	integral	integral	ADJ
ajird-398	228	11	:	:	PUNCT
ajird-398	228	12	𝐼(𝑝	𝐼(𝑝	X
ajird-398	228	13	)	)	PUNCT
ajird-398	228	14	=	=	SYM
ajird-398	228	15	∫	∫	PROPN
ajird-398	228	16	𝑒−𝛼𝑥𝑝	𝑒−𝛼𝑥𝑝	VERB
ajird-398	228	17	𝑥2+𝑏𝑥+𝑐	𝑥2+𝑏𝑥+𝑐	PROPN
ajird-398	228	18	∞	∞	PROPN
ajird-398	228	19	0	0	NUM
ajird-398	228	20	𝑑𝑥.	𝑑𝑥.	NOUN
ajird-398	228	21	solution	solution	NOUN
ajird-398	228	22	.	.	PUNCT
ajird-398	229	1	it	it	PRON
ajird-398	229	2	is	be	AUX
ajird-398	229	3	known	know	VERB
ajird-398	229	4	that	that	SCONJ
ajird-398	229	5	the	the	DET
ajird-398	229	6	given	give	VERB
ajird-398	229	7	integral	integral	ADJ
ajird-398	229	8	depending	depending	NOUN
ajird-398	229	9	on	on	ADP
ajird-398	229	10	the	the	DET
ajird-398	229	11	parameter	parameter	NOUN
ajird-398	229	12	is	be	AUX
ajird-398	229	13	uniformly	uniformly	ADV
ajird-398	229	14	convergent	convergent	NOUN
ajird-398	229	15	for	for	ADP
ajird-398	229	16	𝑎	𝑎	PROPN
ajird-398	229	17	>	>	X
ajird-398	229	18	0	0	PUNCT
ajird-398	230	1	according	accord	VERB
ajird-398	230	2	to	to	ADP
ajird-398	230	3	the	the	DET
ajird-398	230	4	comparison	comparison	NOUN
ajird-398	230	5	property	property	NOUN
ajird-398	230	6	.	.	PUNCT
ajird-398	231	1	so	so	ADV
ajird-398	231	2	,	,	PUNCT
ajird-398	231	3	the	the	DET
ajird-398	231	4	sign	sign	NOUN
ajird-398	231	5	of	of	ADP
ajird-398	231	6	integral	integral	ADJ
ajird-398	231	7	can	can	AUX
ajird-398	231	8	be	be	AUX
ajird-398	231	9	replaced	replace	VERB
ajird-398	231	10	by	by	ADP
ajird-398	231	11	the	the	DET
ajird-398	231	12	sign	sign	NOUN
ajird-398	231	13	of	of	ADP
ajird-398	231	14	sum	sum	NOUN
ajird-398	231	15	.	.	PUNCT
ajird-398	232	1	to	to	PART
ajird-398	232	2	calculate	calculate	VERB
ajird-398	232	3	the	the	DET
ajird-398	232	4	value	value	NOUN
ajird-398	232	5	of	of	ADP
ajird-398	232	6	this	this	DET
ajird-398	232	7	integral	integral	ADJ
ajird-398	232	8	,	,	PUNCT
ajird-398	232	9	we	we	PRON
ajird-398	232	10	use	use	VERB
ajird-398	232	11	the	the	DET
ajird-398	232	12	following	follow	VERB
ajird-398	232	13	taylor	taylor	PROPN
ajird-398	232	14	expansion	expansion	NOUN
ajird-398	232	15	:	:	PUNCT
ajird-398	232	16	1	1	NUM
ajird-398	232	17	𝑥2+𝑏𝑥+𝑐	𝑥2+𝑏𝑥+𝑐	NOUN
ajird-398	232	18	=	=	SYM
ajird-398	232	19	1	1	NUM
ajird-398	232	20	√𝑐	√𝑐	NOUN
ajird-398	232	21	∑	∑	ADP
ajird-398	232	22	𝑥𝑛	𝑥𝑛	PROPN
ajird-398	232	23	sin(𝑛+1)𝜙	sin(𝑛+1)𝜙	PROPN
ajird-398	232	24	sin𝜙	sin𝜙	NOUN
ajird-398	232	25	∞	∞	NUM
ajird-398	232	26	𝑛=0	𝑛=0	PROPN
ajird-398	232	27	,	,	PUNCT
ajird-398	232	28	here	here	ADV
ajird-398	232	29	𝜙	𝜙	NOUN
ajird-398	232	30	=	=	PUNCT
ajird-398	232	31	−arctan√	−arctan√	NOUN
ajird-398	232	32	4𝑐	4𝑐	NOUN
ajird-398	232	33	𝑏2	𝑏2	NOUN
ajird-398	232	34	−	−	PROPN
ajird-398	232	35	1	1	NUM
ajird-398	232	36	.	.	PUNCT
ajird-398	232	37	∫	∫	PROPN
ajird-398	232	38	𝑒−𝛼𝑥𝑝	𝑒−𝛼𝑥𝑝	PROPN
ajird-398	232	39	𝑥2	𝑥2	PROPN
ajird-398	232	40	+	+	CCONJ
ajird-398	232	41	𝑏𝑥	𝑏𝑥	PROPN
ajird-398	232	42	+	+	CCONJ
ajird-398	232	43	𝑐	𝑐	NOUN
ajird-398	232	44	∞	∞	NUM
ajird-398	232	45	0	0	NUM
ajird-398	232	46	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	232	47	=	=	SYM
ajird-398	232	48	∫	∫	PROPN
ajird-398	232	49	𝑒−𝛼𝑥𝑝	𝑒−𝛼𝑥𝑝	VERB
ajird-398	232	50	∑	∑	ADP
ajird-398	232	51	𝑥𝑛	𝑥𝑛	PROPN
ajird-398	232	52	sin(𝑛	sin(𝑛	PROPN
ajird-398	232	53	+	+	CCONJ
ajird-398	232	54	1)𝜙	1)𝜙	PROPN
ajird-398	232	55	𝑐	𝑐	VERB
ajird-398	232	56	∙	∙	PROPN
ajird-398	232	57	sin𝜙	sin𝜙	NOUN
ajird-398	232	58	∞	∞	NUM
ajird-398	232	59	𝑛=0	𝑛=0	PROPN
ajird-398	233	1	+	+	NOUN
ajird-398	233	2	∞	∞	NOUN
ajird-398	233	3	0	0	NUM
ajird-398	234	1	=	=	SYM
ajird-398	234	2	1	1	NUM
ajird-398	234	3	𝑐	𝑐	NOUN
ajird-398	234	4	∑	∑	NUM
ajird-398	234	5	sin(𝑛	sin(𝑛	PROPN
ajird-398	234	6	+	+	CCONJ
ajird-398	234	7	1)𝜙	1)𝜙	NUM
ajird-398	234	8	sin𝜙	sin𝜙	NOUN
ajird-398	234	9	∫	∫	PROPN
ajird-398	234	10	𝑒−𝛼𝑥𝑝	𝑒−𝛼𝑥𝑝	PROPN
ajird-398	234	11	𝑥𝑛	𝑥𝑛	VERB
ajird-398	234	12	+	+	PROPN
ajird-398	234	13	∞	∞	NOUN
ajird-398	234	14	0	0	NUM
ajird-398	234	15	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	234	16	∞	∞	NUM
ajird-398	234	17	𝑛=0	𝑛=0	PROPN
ajird-398	235	1	=	=	PUNCT
ajird-398	236	1	[	[	PUNCT
ajird-398	236	2	𝛼𝑥𝑝	𝛼𝑥𝑝	NOUN
ajird-398	236	3	=	=	PUNCT
ajird-398	236	4	𝜉	𝜉	NOUN
ajird-398	236	5	𝑥	𝑥	NOUN
ajird-398	236	6	=	=	SYM
ajird-398	236	7	√	√	PROPN
ajird-398	236	8	(	(	PUNCT
ajird-398	236	9	𝜉	𝜉	NOUN
ajird-398	236	10	𝛼	𝛼	NOUN
ajird-398	236	11	)	)	PUNCT
ajird-398	236	12	𝑝	𝑝	NOUN
ajird-398	236	13	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	236	14	=	=	SYM
ajird-398	236	15	𝜉	𝜉	ADP
ajird-398	236	16	1	1	NUM
ajird-398	236	17	𝑝	𝑝	NOUN
ajird-398	236	18	−1	−1	NOUN
ajird-398	236	19	𝑝𝛼	𝑝𝛼	ADP
ajird-398	236	20	1	1	NUM
ajird-398	236	21	𝑝	𝑝	NOUN
ajird-398	236	22	𝑑𝜉	𝑑𝜉	ADP
ajird-398	236	23	]	]	PUNCT
ajird-398	236	24	=	=	SYM
ajird-398	236	25	1	1	NUM
ajird-398	236	26	𝑐	𝑐	NOUN
ajird-398	236	27	∑	∑	NUM
ajird-398	236	28	sin(𝑛	sin(𝑛	PROPN
ajird-398	236	29	+	+	CCONJ
ajird-398	236	30	1)𝜙	1)𝜙	NUM
ajird-398	236	31	sin𝜙	sin𝜙	NOUN
ajird-398	236	32	∫	∫	PROPN
ajird-398	236	33	𝑒−𝜉	𝑒−𝜉	PROPN
ajird-398	236	34	∙	∙	PROPN
ajird-398	236	35	(	(	PUNCT
ajird-398	236	36	𝜉	𝜉	X
ajird-398	236	37	𝛼	𝛼	X
ajird-398	236	38	)	)	PUNCT
ajird-398	236	39	𝑛	𝑛	PRON
ajird-398	236	40	𝑝	𝑝	NOUN
ajird-398	236	41	∙	∙	PROPN
ajird-398	236	42	𝜉	𝜉	ADP
ajird-398	236	43	1	1	NUM
ajird-398	236	44	𝑝	𝑝	NOUN
ajird-398	236	45	−1	−1	NOUN
ajird-398	236	46	𝑝𝛼	𝑝𝛼	ADP
ajird-398	236	47	1	1	NUM
ajird-398	236	48	𝑝	𝑝	PROPN
ajird-398	236	49	+	+	PROPN
ajird-398	236	50	∞	∞	NOUN
ajird-398	236	51	0	0	NUM
ajird-398	236	52	𝑑𝜉	𝑑𝜉	ADP
ajird-398	236	53	∞	∞	NUM
ajird-398	236	54	𝑛=0	𝑛=0	PROPN
ajird-398	237	1	=	=	PUNCT
ajird-398	237	2	=	=	SYM
ajird-398	237	3	1	1	NUM
ajird-398	237	4	𝑐	𝑐	NOUN
ajird-398	237	5	∑	∑	NUM
ajird-398	237	6	sin(𝑛	sin(𝑛	PROPN
ajird-398	237	7	+	+	CCONJ
ajird-398	237	8	1)𝜙	1)𝜙	PROPN
ajird-398	237	9	𝑝	𝑝	PROPN
ajird-398	237	10	∙	∙	PROPN
ajird-398	237	11	𝛼	𝛼	NOUN
ajird-398	237	12	𝑛+1	𝑛+1	PROPN
ajird-398	237	13	𝑝	𝑝	PROPN
ajird-398	237	14	∙	∙	PROPN
ajird-398	237	15	sin𝜙	sin𝜙	NOUN
ajird-398	237	16	∫	∫	PROPN
ajird-398	237	17	𝑒−𝜉𝜉	𝑒−𝜉𝜉	PROPN
ajird-398	238	1	𝑛+1	𝑛+1	PROPN
ajird-398	238	2	𝑝	𝑝	PROPN
ajird-398	238	3	−1	−1	NOUN
ajird-398	239	1	+	+	PROPN
ajird-398	239	2	∞	∞	NOUN
ajird-398	239	3	0	0	NUM
ajird-398	239	4	∞	∞	NUM
ajird-398	239	5	𝑛=0	𝑛=0	PROPN
ajird-398	239	6	𝑑𝜉	𝑑𝜉	ADP
ajird-398	239	7	=	=	SYM
ajird-398	239	8	1	1	NUM
ajird-398	239	9	𝑐	𝑐	NOUN
ajird-398	239	10	∑	∑	NUM
ajird-398	239	11	sin(𝑛	sin(𝑛	PROPN
ajird-398	239	12	+	+	CCONJ
ajird-398	239	13	1)𝜙	1)𝜙	PROPN
ajird-398	239	14	𝑝	𝑝	AUX
ajird-398	239	15	∙	∙	PROPN
ajird-398	239	16	𝛼	𝛼	NOUN
ajird-398	239	17	𝑛+1	𝑛+1	PROPN
ajird-398	239	18	𝑝	𝑝	PROPN
ajird-398	239	19	∙	∙	PROPN
ajird-398	239	20	sin	sin	NOUN
ajird-398	239	21	𝜙	𝜙	PROPN
ajird-398	239	22	г	г	PROPN
ajird-398	239	23	(	(	PUNCT
ajird-398	239	24	𝑛	𝑛	PROPN
ajird-398	239	25	+	+	SYM
ajird-398	239	26	1	1	NUM
ajird-398	239	27	𝑝	𝑝	NOUN
ajird-398	239	28	)	)	PUNCT
ajird-398	239	29	∞	∞	NUM
ajird-398	239	30	𝑛=0	𝑛=0	PROPN
ajird-398	239	31	.	.	PUNCT
ajird-398	240	1	problem	problem	NOUN
ajird-398	240	2	10	10	NUM
ajird-398	240	3	.	.	PUNCT
ajird-398	241	1	evaluate	evaluate	VERB
ajird-398	241	2	the	the	DET
ajird-398	241	3	value	value	NOUN
ajird-398	241	4	of	of	ADP
ajird-398	241	5	the	the	DET
ajird-398	241	6	following	follow	VERB
ajird-398	241	7	integral	integral	ADJ
ajird-398	241	8	:	:	PUNCT
ajird-398	241	9	𝐸(𝑘	𝐸(𝑘	NOUN
ajird-398	241	10	)	)	PUNCT
ajird-398	241	11	=	=	SYM
ajird-398	242	1	∫	∫	PROPN
ajird-398	242	2	√1	√1	PART
ajird-398	242	3	−	−	PROPN
ajird-398	242	4	𝑘2	𝑘2	PROPN
ajird-398	242	5	sin2	sin2	NOUN
ajird-398	242	6	𝜙	𝜙	NOUN
ajird-398	242	7	𝜋	𝜋	NOUN
ajird-398	242	8	2	2	NUM
ajird-398	242	9	0	0	NUM
ajird-398	242	10	𝑑𝜙.	𝑑𝜙.	NOUN
ajird-398	242	11	solution	solution	NOUN
ajird-398	242	12	.	.	PUNCT
ajird-398	243	1	to	to	PART
ajird-398	243	2	evaluate	evaluate	VERB
ajird-398	243	3	this	this	DET
ajird-398	243	4	integral	integral	ADJ
ajird-398	243	5	,	,	PUNCT
ajird-398	243	6	we	we	PRON
ajird-398	243	7	use	use	VERB
ajird-398	243	8	the	the	DET
ajird-398	243	9	following	follow	VERB
ajird-398	243	10	taylor	taylor	PROPN
ajird-398	243	11	expansion	expansion	NOUN
ajird-398	243	12	√1	√1	ADV
ajird-398	243	13	−	−	PROPN
ajird-398	244	1	𝑥	𝑥	NOUN
ajird-398	244	2	=	=	SYM
ajird-398	244	3	1	1	NUM
ajird-398	244	4	−	−	NOUN
ajird-398	244	5	∑	∑	PUNCT
ajird-398	244	6	(	(	PUNCT
ajird-398	244	7	2𝑛−3)‼	2𝑛−3)‼	NOUN
ajird-398	244	8	2𝑛∙𝑛	2𝑛∙𝑛	NUM
ajird-398	244	9	!	!	PUNCT
ajird-398	245	1	𝑥𝑛∞	𝑥𝑛∞	PROPN
ajird-398	245	2	𝑛=1	𝑛=1	NOUN
ajird-398	245	3	.	.	PUNCT
ajird-398	246	1	in	in	ADP
ajird-398	246	2	that	that	DET
ajird-398	246	3	case	case	NOUN
ajird-398	246	4	,	,	PUNCT
ajird-398	246	5	𝐸(𝑘	𝐸(𝑘	NOUN
ajird-398	246	6	)	)	PUNCT
ajird-398	246	7	=	=	PUNCT
ajird-398	247	1	∫	∫	PROPN
ajird-398	247	2	√1	√1	PART
ajird-398	247	3	−	−	PROPN
ajird-398	247	4	𝑘2	𝑘2	PROPN
ajird-398	247	5	sin2	sin2	NOUN
ajird-398	247	6	𝜙	𝜙	NOUN
ajird-398	247	7	𝜋	𝜋	NOUN
ajird-398	247	8	2	2	NUM
ajird-398	247	9	0	0	NUM
ajird-398	247	10	𝑑𝜙	𝑑𝜙	ADP
ajird-398	247	11	=	=	SYM
ajird-398	247	12	∫(1	∫(1	PROPN
ajird-398	248	1	−	−	NOUN
ajird-398	248	2	∑	∑	PUNCT
ajird-398	248	3	(	(	PUNCT
ajird-398	248	4	(	(	PUNCT
ajird-398	248	5	2𝑛	2𝑛	PROPN
ajird-398	248	6	−	−	PROPN
ajird-398	248	7	3)‼	3)‼	NOUN
ajird-398	248	8	)	)	PUNCT
ajird-398	248	9	2𝑛𝑛	2𝑛𝑛	NOUN
ajird-398	248	10	!	!	PUNCT
ajird-398	249	1	∙	∙	PROPN
ajird-398	249	2	(	(	PUNCT
ajird-398	249	3	𝑘2	𝑘2	PROPN
ajird-398	249	4	sin2	sin2	NOUN
ajird-398	249	5	𝜙)𝑛	𝜙)𝑛	PUNCT
ajird-398	250	1	∞	∞	NUM
ajird-398	250	2	𝑛=1	𝑛=1	NOUN
ajird-398	250	3	𝜋	𝜋	NOUN
ajird-398	250	4	2	2	NUM
ajird-398	250	5	0	0	NUM
ajird-398	250	6	𝑑𝜙	𝑑𝜙	ADP
ajird-398	250	7	=	=	PUNCT
ajird-398	251	1	=	=	PUNCT
ajird-398	251	2	𝜋	𝜋	NOUN
ajird-398	251	3	2	2	NUM
ajird-398	251	4	−	−	NOUN
ajird-398	251	5	∑	∑	PUNCT
ajird-398	251	6	(	(	PUNCT
ajird-398	251	7	2𝑛	2𝑛	NOUN
ajird-398	251	8	−	−	NOUN
ajird-398	252	1	3)‼	3)‼	NOUN
ajird-398	252	2	𝑘2𝑛	𝑘2𝑛	PROPN
ajird-398	252	3	2𝑛𝑛	2𝑛𝑛	NOUN
ajird-398	252	4	!	!	PUNCT
ajird-398	253	1	∫	∫	PROPN
ajird-398	253	2	sin2𝑛	sin2𝑛	PART
ajird-398	254	1	𝜙	𝜙	NOUN
ajird-398	254	2	𝜋	𝜋	NOUN
ajird-398	254	3	2	2	NUM
ajird-398	254	4	0	0	NUM
ajird-398	254	5	𝑑𝜙	𝑑𝜙	NUM
ajird-398	254	6	∞	∞	NUM
ajird-398	254	7	𝑛=1	𝑛=1	NOUN
ajird-398	254	8	=	=	PRON
ajird-398	254	9	[	[	PUNCT
ajird-398	254	10	∫	∫	PROPN
ajird-398	254	11	sin2𝑛	sin2𝑛	PROPN
ajird-398	254	12	𝜙	𝜙	PROPN
ajird-398	254	13	𝜋	𝜋	NOUN
ajird-398	254	14	2	2	NUM
ajird-398	254	15	0	0	NUM
ajird-398	254	16	𝑑𝜙	𝑑𝜙	NOUN
ajird-398	254	17	=	=	SYM
ajird-398	254	18	(	(	PUNCT
ajird-398	254	19	2𝑛	2𝑛	PROPN
ajird-398	254	20	−	−	PROPN
ajird-398	254	21	1)‼	1)‼	PROPN
ajird-398	254	22	(	(	PUNCT
ajird-398	254	23	2𝑛)‼	2𝑛)‼	NUM
ajird-398	254	24	∙	∙	PROPN
ajird-398	254	25	𝜋	𝜋	ADP
ajird-398	254	26	2	2	NUM
ajird-398	254	27	]	]	PUNCT
ajird-398	254	28	=	=	SYM
ajird-398	254	29	american	american	PROPN
ajird-398	254	30	journal	journal	PROPN
ajird-398	254	31	of	of	ADP
ajird-398	254	32	interdisciplinary	interdisciplinary	ADJ
ajird-398	254	33	research	research	NOUN
ajird-398	254	34	and	and	CCONJ
ajird-398	254	35	development	development	NOUN
ajird-398	254	36	issn	issn	PROPN
ajird-398	254	37	online	online	NOUN
ajird-398	254	38	:	:	PUNCT
ajird-398	254	39	2771	2771	NUM
ajird-398	254	40	-	-	SYM
ajird-398	254	41	8948	8948	NUM
ajird-398	254	42	website	website	NOUN
ajird-398	254	43	:	:	PUNCT
ajird-398	254	44	www.ajird.journalspark.org	www.ajird.journalspark.org	ADJ
ajird-398	254	45	volume	volume	NOUN
ajird-398	254	46	11	11	NUM
ajird-398	254	47	,	,	PUNCT
ajird-398	254	48	dec	dec	PROPN
ajird-398	254	49	.	.	PROPN
ajird-398	254	50	,	,	PUNCT
ajird-398	254	51	2022	2022	NUM
ajird-398	254	52	43	43	NUM
ajird-398	255	1	|	|	ADV
ajird-398	255	2	p	p	X
ajird-398	255	3	a	a	DET
ajird-398	255	4	g	g	NOUN
ajird-398	255	5	e	e	NOUN
ajird-398	255	6	=	=	NOUN
ajird-398	255	7	𝜋	𝜋	X
ajird-398	255	8	2	2	NUM
ajird-398	255	9	−	−	NOUN
ajird-398	255	10	∑	∑	PUNCT
ajird-398	255	11	(	(	PUNCT
ajird-398	255	12	2𝑛	2𝑛	NOUN
ajird-398	255	13	−	−	NOUN
ajird-398	256	1	3)‼	3)‼	NOUN
ajird-398	256	2	𝑘2𝑛	𝑘2𝑛	NUM
ajird-398	256	3	2𝑛𝑛	2𝑛𝑛	NOUN
ajird-398	256	4	!	!	PUNCT
ajird-398	257	1	∙	∙	PROPN
ajird-398	257	2	(	(	PUNCT
ajird-398	257	3	2𝑛	2𝑛	PROPN
ajird-398	257	4	−	−	PROPN
ajird-398	257	5	1)‼	1)‼	PROPN
ajird-398	257	6	(	(	PUNCT
ajird-398	257	7	2𝑛)‼	2𝑛)‼	NUM
ajird-398	257	8	∙	∙	PROPN
ajird-398	257	9	𝜋	𝜋	ADP
ajird-398	257	10	2	2	NUM
ajird-398	257	11	∞	∞	NUM
ajird-398	257	12	𝑛=1	𝑛=1	NOUN
ajird-398	257	13	=	=	NOUN
ajird-398	257	14	𝜋	𝜋	X
ajird-398	257	15	2	2	NUM
ajird-398	257	16	(	(	PUNCT
ajird-398	257	17	1	1	NUM
ajird-398	257	18	−	−	NOUN
ajird-398	257	19	∑	∑	PUNCT
ajird-398	257	20	(	(	PUNCT
ajird-398	257	21	(	(	PUNCT
ajird-398	257	22	2𝑛	2𝑛	PROPN
ajird-398	257	23	−	−	PROPN
ajird-398	257	24	1)‼	1)‼	PROPN
ajird-398	257	25	𝑛	𝑛	PROPN
ajird-398	257	26	!	!	PUNCT
ajird-398	257	27	)	)	PUNCT
ajird-398	258	1	2	2	NUM
ajird-398	258	2	∙	∙	PROPN
ajird-398	258	3	𝑘2𝑛	𝑘2𝑛	NUM
ajird-398	258	4	22𝑛(2𝑛	22𝑛(2𝑛	NUM
ajird-398	258	5	−	−	NOUN
ajird-398	258	6	1	1	NUM
ajird-398	258	7	)	)	PUNCT
ajird-398	258	8	∞	∞	NUM
ajird-398	258	9	𝑛=1	𝑛=1	NOUN
ajird-398	258	10	)	)	PUNCT
ajird-398	258	11	=	=	PUNCT
ajird-398	259	1	=	=	PUNCT
ajird-398	259	2	𝜋	𝜋	X
ajird-398	259	3	2	2	NUM
ajird-398	259	4	(	(	PUNCT
ajird-398	259	5	1	1	NUM
ajird-398	259	6	−	−	NOUN
ajird-398	259	7	∑	∑	PUNCT
ajird-398	259	8	(	(	PUNCT
ajird-398	259	9	(	(	PUNCT
ajird-398	259	10	2𝑛	2𝑛	PROPN
ajird-398	259	11	−	−	PROPN
ajird-398	259	12	1)‼	1)‼	PROPN
ajird-398	259	13	(	(	PUNCT
ajird-398	259	14	2𝑛	2𝑛	NUM
ajird-398	259	15	)	)	PUNCT
ajird-398	259	16	!	!	PUNCT
ajird-398	259	17	!	!	PUNCT
ajird-398	259	18	)	)	PUNCT
ajird-398	260	1	2	2	NUM
ajird-398	260	2	∙	∙	PROPN
ajird-398	260	3	𝑘2𝑛	𝑘2𝑛	NUM
ajird-398	260	4	(	(	PUNCT
ajird-398	260	5	2𝑛	2𝑛	PROPN
ajird-398	260	6	−	−	PROPN
ajird-398	260	7	1	1	NUM
ajird-398	260	8	)	)	PUNCT
ajird-398	260	9	∞	∞	NUM
ajird-398	260	10	𝑛=1	𝑛=1	NOUN
ajird-398	260	11	)	)	PUNCT
ajird-398	260	12	.	.	PUNCT
ajird-398	261	1	if	if	SCONJ
ajird-398	261	2	we	we	PRON
ajird-398	261	3	take	take	VERB
ajird-398	261	4	𝑘	𝑘	PRON
ajird-398	261	5	=	=	NOUN
ajird-398	261	6	1	1	NUM
ajird-398	261	7	2	2	NUM
ajird-398	261	8	in	in	ADP
ajird-398	261	9	the	the	DET
ajird-398	261	10	given	give	VERB
ajird-398	261	11	integral	integral	ADJ
ajird-398	261	12	,	,	PUNCT
ajird-398	261	13	the	the	DET
ajird-398	261	14	following	follow	VERB
ajird-398	261	15	equality	equality	NOUN
ajird-398	261	16	is	be	AUX
ajird-398	261	17	holds	hold	VERB
ajird-398	261	18	:	:	PUNCT
ajird-398	261	19	∫	∫	PROPN
ajird-398	261	20	√1	√1	ADV
ajird-398	261	21	−	−	PROPN
ajird-398	261	22	1	1	NUM
ajird-398	261	23	4	4	NUM
ajird-398	261	24	sin2	sin2	NOUN
ajird-398	261	25	𝑥	𝑥	NOUN
ajird-398	261	26	𝜋	𝜋	NOUN
ajird-398	261	27	2	2	NUM
ajird-398	261	28	0	0	NUM
ajird-398	261	29	𝑑𝑥	𝑑𝑥	NOUN
ajird-398	261	30	=	=	SYM
ajird-398	261	31	𝜋	𝜋	NOUN
ajird-398	261	32	2	2	NUM
ajird-398	261	33	(	(	PUNCT
ajird-398	261	34	1	1	NUM
ajird-398	261	35	−	−	NOUN
ajird-398	261	36	∑	∑	PUNCT
ajird-398	261	37	(	(	PUNCT
ajird-398	261	38	(	(	PUNCT
ajird-398	261	39	2𝑛	2𝑛	PROPN
ajird-398	261	40	−	−	PROPN
ajird-398	261	41	1)‼	1)‼	PROPN
ajird-398	261	42	(	(	PUNCT
ajird-398	261	43	2𝑛	2𝑛	NUM
ajird-398	261	44	)	)	PUNCT
ajird-398	261	45	!	!	PUNCT
ajird-398	261	46	!	!	PUNCT
ajird-398	261	47	)	)	PUNCT
ajird-398	262	1	2	2	NUM
ajird-398	262	2	∙	∙	NOUN
ajird-398	262	3	1	1	NUM
ajird-398	262	4	22𝑛(2𝑛	22𝑛(2𝑛	NUM
ajird-398	262	5	−	−	NOUN
ajird-398	262	6	1	1	NUM
ajird-398	262	7	)	)	PUNCT
ajird-398	262	8	∞	∞	NUM
ajird-398	262	9	𝑛=1	𝑛=1	NOUN
ajird-398	262	10	)	)	PUNCT
ajird-398	262	11	.	.	PUNCT
ajird-398	263	1	problem	problem	NOUN
ajird-398	263	2	11	11	NUM
ajird-398	263	3	.	.	PUNCT
ajird-398	263	4	evaluate	evaluate	VERB
ajird-398	263	5	the	the	DET
ajird-398	263	6	value	value	NOUN
ajird-398	263	7	of	of	ADP
ajird-398	263	8	the	the	DET
ajird-398	263	9	bessel	bessel	NOUN
ajird-398	263	10	function	function	NOUN
ajird-398	263	11	:	:	PUNCT
ajird-398	263	12	𝐼(𝑥	𝐼(𝑥	X
ajird-398	263	13	)	)	PUNCT
ajird-398	263	14	=	=	SYM
ajird-398	263	15	1	1	NUM
ajird-398	263	16	𝜋	𝜋	NOUN
ajird-398	263	17	∫	∫	PROPN
ajird-398	263	18	cos(𝑥	cos(𝑥	PROPN
ajird-398	263	19	∙	∙	PROPN
ajird-398	263	20	sin𝜙	sin𝜙	NOUN
ajird-398	263	21	)	)	PUNCT
ajird-398	263	22	𝜋	𝜋	NOUN
ajird-398	263	23	0	0	NUM
ajird-398	263	24	𝑑𝜙.	𝑑𝜙.	NOUN
ajird-398	263	25	solution	solution	NOUN
ajird-398	263	26	.	.	PUNCT
ajird-398	264	1	to	to	PART
ajird-398	264	2	evaluate	evaluate	VERB
ajird-398	264	3	this	this	DET
ajird-398	264	4	integral	integral	ADJ
ajird-398	264	5	,	,	PUNCT
ajird-398	264	6	we	we	PRON
ajird-398	264	7	use	use	VERB
ajird-398	264	8	the	the	DET
ajird-398	264	9	following	follow	VERB
ajird-398	264	10	taylor	taylor	PROPN
ajird-398	264	11	expansion	expansion	NOUN
ajird-398	264	12	cos	cos	ADP
ajird-398	264	13	𝑥	𝑥	PROPN
ajird-398	264	14	=	=	SYM
ajird-398	264	15	∑	∑	PROPN
ajird-398	264	16	(	(	PUNCT
ajird-398	264	17	(	(	PUNCT
ajird-398	264	18	−1)𝑛𝑥2𝑛	−1)𝑛𝑥2𝑛	NOUN
ajird-398	264	19	)	)	PUNCT
ajird-398	264	20	(	(	PUNCT
ajird-398	264	21	2𝑛	2𝑛	NUM
ajird-398	264	22	)	)	PUNCT
ajird-398	264	23	!	!	PUNCT
ajird-398	265	1	∞	∞	NUM
ajird-398	265	2	𝑛=0	𝑛=0	PROPN
ajird-398	265	3	.	.	PUNCT
ajird-398	266	1	in	in	ADP
ajird-398	266	2	that	that	DET
ajird-398	266	3	case	case	NOUN
ajird-398	266	4	,	,	PUNCT
ajird-398	266	5	𝐼(𝑥	𝐼(𝑥	PROPN
ajird-398	266	6	)	)	PUNCT
ajird-398	266	7	=	=	SYM
ajird-398	266	8	1	1	NUM
ajird-398	266	9	𝜋	𝜋	NOUN
ajird-398	266	10	∫	∫	PROPN
ajird-398	266	11	cos(𝑥	cos(𝑥	PROPN
ajird-398	266	12	∙	∙	PROPN
ajird-398	266	13	sin	sin	PROPN
ajird-398	266	14	𝜙	𝜙	NOUN
ajird-398	266	15	)	)	PUNCT
ajird-398	266	16	𝜋	𝜋	NOUN
ajird-398	266	17	0	0	NUM
ajird-398	266	18	𝑑𝜙	𝑑𝜙	VERB
ajird-398	266	19	=	=	SYM
ajird-398	266	20	1	1	NUM
ajird-398	266	21	𝜋	𝜋	NOUN
ajird-398	266	22	∫	∫	PROPN
ajird-398	266	23	∑	∑	INTJ
ajird-398	266	24	(	(	PUNCT
ajird-398	266	25	−1)𝑛(𝑥	−1)𝑛(𝑥	X
ajird-398	266	26	∙	∙	PROPN
ajird-398	266	27	sin𝜙)2𝑛	sin𝜙)2𝑛	PROPN
ajird-398	266	28	(	(	PUNCT
ajird-398	266	29	2𝑛	2𝑛	NUM
ajird-398	266	30	)	)	PUNCT
ajird-398	266	31	!	!	PUNCT
ajird-398	267	1	∞	∞	NUM
ajird-398	268	1	𝑛=0	𝑛=0	PROPN
ajird-398	269	1	𝜋	𝜋	NOUN
ajird-398	269	2	0	0	NUM
ajird-398	269	3	𝑑𝜙	𝑑𝜙	ADP
ajird-398	269	4	=	=	SYM
ajird-398	270	1	=	=	SYM
ajird-398	270	2	1	1	NUM
ajird-398	270	3	𝜋	𝜋	NOUN
ajird-398	270	4	∑	∑	PROPN
ajird-398	270	5	(	(	PUNCT
ajird-398	270	6	−1)𝑛𝑥2𝑛	−1)𝑛𝑥2𝑛	PROPN
ajird-398	270	7	(	(	PUNCT
ajird-398	270	8	2𝑛	2𝑛	NUM
ajird-398	270	9	)	)	PUNCT
ajird-398	270	10	!	!	PUNCT
ajird-398	271	1	∫	∫	PROPN
ajird-398	272	1	sin2n	sin2n	PROPN
ajird-398	272	2	𝜙	𝜙	PROPN
ajird-398	273	1	𝜋	𝜋	NOUN
ajird-398	273	2	0	0	NUM
ajird-398	273	3	𝑑𝜙	𝑑𝜙	ADP
ajird-398	273	4	∞	∞	NUM
ajird-398	273	5	𝑛=0	𝑛=0	PROPN
ajird-398	273	6	=	=	PUNCT
ajird-398	273	7	1	1	NUM
ajird-398	273	8	𝜋	𝜋	NOUN
ajird-398	273	9	∑	∑	PROPN
ajird-398	273	10	(	(	PUNCT
ajird-398	273	11	−1)𝑛𝑥2𝑛	−1)𝑛𝑥2𝑛	PROPN
ajird-398	273	12	(	(	PUNCT
ajird-398	273	13	2𝑛	2𝑛	NUM
ajird-398	273	14	)	)	PUNCT
ajird-398	273	15	!	!	PUNCT
ajird-398	274	1	∫	∫	PROPN
ajird-398	275	1	sin2n	sin2n	PROPN
ajird-398	276	1	𝜙	𝜙	PROPN
ajird-398	276	2	𝜋	𝜋	NOUN
ajird-398	276	3	2	2	NUM
ajird-398	276	4	0	0	NUM
ajird-398	276	5	𝑑𝜙	𝑑𝜙	ADP
ajird-398	276	6	∞	∞	NUM
ajird-398	276	7	𝑛=0	𝑛=0	PROPN
ajird-398	277	1	+	+	CCONJ
ajird-398	277	2	1	1	NUM
ajird-398	277	3	𝜋	𝜋	NOUN
ajird-398	277	4	∑	∑	PROPN
ajird-398	277	5	(	(	PUNCT
ajird-398	277	6	−1)𝑛𝑥2𝑛	−1)𝑛𝑥2𝑛	PROPN
ajird-398	277	7	(	(	PUNCT
ajird-398	277	8	2𝑛	2𝑛	NUM
ajird-398	277	9	)	)	PUNCT
ajird-398	277	10	!	!	PUNCT
ajird-398	278	1	∫	∫	PROPN
ajird-398	278	2	sin2𝑛	sin2𝑛	PRON
ajird-398	279	1	𝜙	𝜙	PRON
ajird-398	279	2	𝜋	𝜋	NOUN
ajird-398	279	3	𝜋	𝜋	NOUN
ajird-398	279	4	2	2	NUM
ajird-398	279	5	𝑑𝜙	𝑑𝜙	ADP
ajird-398	279	6	∞	∞	NUM
ajird-398	279	7	𝑛=0	𝑛=0	PROPN
ajird-398	279	8	.	.	PUNCT
ajird-398	280	1	in	in	ADP
ajird-398	280	2	the	the	DET
ajird-398	280	3	second	second	ADJ
ajird-398	280	4	integral	integral	NOUN
ajird-398	280	5	on	on	ADP
ajird-398	280	6	the	the	DET
ajird-398	280	7	right	right	ADJ
ajird-398	280	8	side	side	NOUN
ajird-398	280	9	of	of	ADP
ajird-398	280	10	the	the	DET
ajird-398	280	11	last	last	ADJ
ajird-398	280	12	equality	equality	NOUN
ajird-398	280	13	,	,	PUNCT
ajird-398	280	14	we	we	PRON
ajird-398	280	15	can	can	AUX
ajird-398	280	16	define	define	VERB
ajird-398	280	17	it	it	PRON
ajird-398	280	18	as	as	ADP
ajird-398	280	19	𝜙	𝜙	PROPN
ajird-398	280	20	=	=	X
ajird-398	280	21	𝜋	𝜋	NOUN
ajird-398	280	22	2	2	NUM
ajird-398	280	23	+	+	NUM
ajird-398	280	24	𝜓.	𝜓.	NOUN
ajird-398	280	25	in	in	ADP
ajird-398	280	26	that	that	DET
ajird-398	280	27	case	case	NOUN
ajird-398	280	28	,	,	PUNCT
ajird-398	280	29	𝐼(𝑥	𝐼(𝑥	PROPN
ajird-398	280	30	)	)	PUNCT
ajird-398	280	31	=	=	SYM
ajird-398	280	32	1	1	NUM
ajird-398	280	33	𝜋	𝜋	NOUN
ajird-398	280	34	∑	∑	PROPN
ajird-398	280	35	(	(	PUNCT
ajird-398	280	36	−1)𝑛𝑥2𝑛	−1)𝑛𝑥2𝑛	PROPN
ajird-398	280	37	(	(	PUNCT
ajird-398	280	38	2𝑛	2𝑛	NUM
ajird-398	280	39	)	)	PUNCT
ajird-398	280	40	!	!	PUNCT
ajird-398	281	1	∫	∫	PROPN
ajird-398	282	1	sin2n	sin2n	PROPN
ajird-398	283	1	𝜙	𝜙	PROPN
ajird-398	283	2	𝜋	𝜋	NOUN
ajird-398	283	3	2	2	NUM
ajird-398	283	4	0	0	NUM
ajird-398	283	5	𝑑𝜙	𝑑𝜙	ADP
ajird-398	283	6	∞	∞	NUM
ajird-398	283	7	𝑛=0	𝑛=0	PROPN
ajird-398	284	1	+	+	CCONJ
ajird-398	284	2	1	1	NUM
ajird-398	284	3	𝜋	𝜋	NOUN
ajird-398	284	4	∑	∑	PROPN
ajird-398	284	5	(	(	PUNCT
ajird-398	284	6	−1)𝑛𝑥2𝑛	−1)𝑛𝑥2𝑛	PROPN
ajird-398	284	7	(	(	PUNCT
ajird-398	284	8	2𝑛	2𝑛	NUM
ajird-398	284	9	)	)	PUNCT
ajird-398	284	10	!	!	PUNCT
ajird-398	285	1	∫	∫	PROPN
ajird-398	286	1	cos2𝑛	cos2𝑛	INTJ
ajird-398	286	2	𝜓	𝜓	PROPN
ajird-398	286	3	𝜋	𝜋	NOUN
ajird-398	286	4	2	2	NUM
ajird-398	286	5	0	0	NUM
ajird-398	286	6	𝑑𝜓	𝑑𝜓	NUM
ajird-398	286	7	∞	∞	NUM
ajird-398	286	8	𝑛=0	𝑛=0	PROPN
ajird-398	287	1	=	=	PUNCT
ajird-398	287	2	=	=	SYM
ajird-398	287	3	2	2	NUM
ajird-398	287	4	𝜋	𝜋	NOUN
ajird-398	287	5	∑	∑	PUNCT
ajird-398	287	6	(	(	PUNCT
ajird-398	287	7	−1)𝑛𝑥2𝑛	−1)𝑛𝑥2𝑛	PROPN
ajird-398	287	8	(	(	PUNCT
ajird-398	287	9	2𝑛	2𝑛	NUM
ajird-398	287	10	)	)	PUNCT
ajird-398	287	11	!	!	PUNCT
ajird-398	288	1	∫	∫	PROPN
ajird-398	289	1	sin2n	sin2n	PROPN
ajird-398	290	1	𝜙	𝜙	PROPN
ajird-398	290	2	𝜋	𝜋	NOUN
ajird-398	290	3	2	2	NUM
ajird-398	290	4	0	0	NUM
ajird-398	290	5	𝑑𝜙	𝑑𝜙	ADP
ajird-398	290	6	∞	∞	NUM
ajird-398	290	7	𝑛=0	𝑛=0	NOUN
ajird-398	290	8	=	=	SYM
ajird-398	291	1	2	2	NUM
ajird-398	291	2	𝜋	𝜋	NOUN
ajird-398	291	3	∑	∑	PUNCT
ajird-398	291	4	(	(	PUNCT
ajird-398	291	5	−1)𝑛𝑥2𝑛	−1)𝑛𝑥2𝑛	PROPN
ajird-398	291	6	(	(	PUNCT
ajird-398	291	7	2𝑛	2𝑛	NUM
ajird-398	291	8	)	)	PUNCT
ajird-398	291	9	!	!	PUNCT
ajird-398	292	1	∙	∙	PROPN
ajird-398	292	2	(	(	PUNCT
ajird-398	292	3	2𝑛	2𝑛	PROPN
ajird-398	292	4	−	−	PROPN
ajird-398	292	5	1)‼	1)‼	PROPN
ajird-398	292	6	(	(	PUNCT
ajird-398	292	7	2𝑛)‼	2𝑛)‼	NUM
ajird-398	292	8	∙	∙	NOUN
ajird-398	292	9	𝜋	𝜋	ADP
ajird-398	292	10	2	2	NUM
ajird-398	292	11	∞	∞	NUM
ajird-398	292	12	𝑛=0	𝑛=0	NOUN
ajird-398	292	13	=	=	PUNCT
ajird-398	293	1	=	=	PUNCT
ajird-398	293	2	∑	∑	PUNCT
ajird-398	293	3	(	(	PUNCT
ajird-398	293	4	−1)𝑛𝑥2𝑛	−1)𝑛𝑥2𝑛	PROPN
ajird-398	293	5	(	(	PUNCT
ajird-398	293	6	2𝑛	2𝑛	NUM
ajird-398	293	7	)	)	PUNCT
ajird-398	293	8	!	!	PUNCT
ajird-398	294	1	∙	∙	PROPN
ajird-398	294	2	(	(	PUNCT
ajird-398	294	3	2𝑛	2𝑛	PROPN
ajird-398	294	4	−	−	PROPN
ajird-398	294	5	1)‼	1)‼	PROPN
ajird-398	294	6	(	(	PUNCT
ajird-398	294	7	2𝑛)‼	2𝑛)‼	NUM
ajird-398	294	8	∞	∞	NUM
ajird-398	294	9	𝑛=0	𝑛=0	PROPN
ajird-398	294	10	=	=	PUNCT
ajird-398	294	11	∑	∑	PROPN
ajird-398	294	12	(	(	PUNCT
ajird-398	294	13	−1)𝑛𝑥2𝑛	−1)𝑛𝑥2𝑛	PROPN
ajird-398	294	14	(	(	PUNCT
ajird-398	294	15	2𝑛	2𝑛	NUM
ajird-398	294	16	)	)	PUNCT
ajird-398	294	17	!	!	PUNCT
ajird-398	295	1	∙	∙	PROPN
ajird-398	295	2	(	(	PUNCT
ajird-398	295	3	2𝑛	2𝑛	PROPN
ajird-398	295	4	−	−	PROPN
ajird-398	295	5	1)‼	1)‼	PROPN
ajird-398	296	1	2𝑛	2𝑛	PROPN
ajird-398	296	2	∙	∙	PROPN
ajird-398	296	3	𝑛	𝑛	PROPN
ajird-398	296	4	!	!	NOUN
ajird-398	296	5	∞	∞	NUM
ajird-398	296	6	𝑛=0	𝑛=0	PROPN
ajird-398	296	7	.	.	PUNCT
ajird-398	297	1	american	american	ADJ
ajird-398	297	2	journal	journal	PROPN
ajird-398	297	3	of	of	ADP
ajird-398	297	4	interdisciplinary	interdisciplinary	ADJ
ajird-398	297	5	research	research	NOUN
ajird-398	297	6	and	and	CCONJ
ajird-398	297	7	development	development	NOUN
ajird-398	297	8	issn	issn	PROPN
ajird-398	297	9	online	online	NOUN
ajird-398	297	10	:	:	PUNCT
ajird-398	297	11	2771	2771	NUM
ajird-398	297	12	-	-	SYM
ajird-398	297	13	8948	8948	NUM
ajird-398	297	14	website	website	NOUN
ajird-398	297	15	:	:	PUNCT
ajird-398	297	16	www.ajird.journalspark.org	www.ajird.journalspark.org	ADJ
ajird-398	297	17	volume	volume	NOUN
ajird-398	297	18	11	11	NUM
ajird-398	297	19	,	,	PUNCT
ajird-398	297	20	dec	dec	PROPN
ajird-398	297	21	.	.	PROPN
ajird-398	297	22	,	,	PUNCT
ajird-398	297	23	2022	2022	NUM
ajird-398	297	24	44	44	NUM
ajird-398	298	1	|	|	ADV
ajird-398	298	2	p	p	NOUN
ajird-398	298	3	a	a	DET
ajird-398	298	4	g	g	NOUN
ajird-398	298	5	e	e	NOUN
ajird-398	298	6	similar	similar	ADJ
ajird-398	298	7	expansions	expansion	NOUN
ajird-398	298	8	can	can	AUX
ajird-398	298	9	be	be	AUX
ajird-398	298	10	successfully	successfully	ADV
ajird-398	298	11	used	use	VERB
ajird-398	298	12	to	to	PART
ajird-398	298	13	approximate	approximate	ADJ
ajird-398	298	14	integrals	integral	NOUN
ajird-398	298	15	that	that	PRON
ajird-398	298	16	can	can	AUX
ajird-398	298	17	not	not	PART
ajird-398	298	18	be	be	AUX
ajird-398	298	19	expressed	express	VERB
ajird-398	298	20	in	in	ADP
ajird-398	298	21	finite	finite	ADJ
ajird-398	298	22	form	form	NOUN
ajird-398	298	23	and	and	CCONJ
ajird-398	298	24	to	to	PART
ajird-398	298	25	compile	compile	VERB
ajird-398	298	26	tables	table	NOUN
ajird-398	298	27	of	of	ADP
ajird-398	298	28	their	their	PRON
ajird-398	298	29	values	value	NOUN
ajird-398	298	30	.	.	PUNCT
ajird-398	299	1	references	reference	NOUN
ajird-398	299	2	1	1	NUM
ajird-398	299	3	.	.	PUNCT
ajird-398	299	4	alimov	alimov	PROPN
ajird-398	299	5	sh	sh	PROPN
ajird-398	299	6	.	.	PROPN
ajird-398	299	7	,	,	PUNCT
ajird-398	299	8	ashurov	ashurov	PROPN
ajird-398	299	9	r.	r.	PROPN
ajird-398	299	10	matematik	matematik	PROPN
ajird-398	299	11	tahlil	tahlil	PROPN
ajird-398	299	12	.	.	PUNCT
ajird-398	300	1	3	3	NUM
ajird-398	300	2	-	-	PUNCT
ajird-398	300	3	qism	qism	NOUN
ajird-398	300	4	.	.	PUNCT
ajird-398	301	1	“	"	PUNCT
ajird-398	301	2	mumtoz	mumtoz	VERB
ajird-398	301	3	so`z	so`z	NOUN
ajird-398	301	4	”	"	PUNCT
ajird-398	301	5	,	,	PUNCT
ajird-398	301	6	toshkent	toshkent	NOUN
ajird-398	301	7	,	,	PUNCT
ajird-398	301	8	2018	2018	NUM
ajird-398	301	9	.	.	PUNCT
ajird-398	302	1	2	2	X
ajird-398	302	2	.	.	X
ajird-398	302	3	демидович	демидович	PROPN
ajird-398	302	4	б.п	б.п	PROPN
ajird-398	302	5	.	.	PROPN
ajird-398	302	6	сборник	сборник	PROPN
ajird-398	302	7	задач	задач	VERB
ajird-398	302	8	и	и	PROPN
ajird-398	302	9	упражнений	упражнений	PROPN
ajird-398	302	10	по	по	PROPN
ajird-398	302	11	математическому	математическому	PROPN
ajird-398	302	12	анализу	анализу	PROPN
ajird-398	302	13	.	.	PUNCT
ajird-398	303	1	издательство	издательство	PROPN
ajird-398	303	2	черо	черо	PROPN
ajird-398	303	3	,	,	PUNCT
ajird-398	303	4	13	13	NUM
ajird-398	303	5	-	-	PUNCT
ajird-398	303	6	е	е	NOUN
ajird-398	303	7	издание	издание	NOUN
ajird-398	303	8	.	.	PUNCT
ajird-398	304	1	1997	1997	NUM
ajird-398	304	2	,	,	PUNCT
ajird-398	304	3	москва	москва	PROPN
ajird-398	304	4	.	.	PUNCT
ajird-398	305	1	3	3	X
ajird-398	305	2	.	.	X
ajird-398	305	3	sa’dullayev	sa’dullayev	PROPN
ajird-398	305	4	a.	a.	PROPN
ajird-398	305	5	,	,	PUNCT
ajird-398	305	6	mansurov	mansurov	PROPN
ajird-398	305	7	h.	h.	PROPN
ajird-398	305	8	,	,	PUNCT
ajird-398	305	9	xudoyberganov	xudoyberganov	PROPN
ajird-398	305	10	g.	g.	PROPN
ajird-398	305	11	va	va	PROPN
ajird-398	305	12	b.q	b.q	PROPN
ajird-398	305	13	.	.	PROPN
ajird-398	305	14	matematik	matematik	PROPN
ajird-398	305	15	analiz	analiz	PROPN
ajird-398	305	16	kursidan	kursidan	PROPN
ajird-398	305	17	misol	misol	PROPN
ajird-398	305	18	va	va	PROPN
ajird-398	305	19	masalalar	masalalar	PROPN
ajird-398	305	20	to`plami	to`plami	PROPN
ajird-398	305	21	.	.	PUNCT
ajird-398	306	1	3	3	NUM
ajird-398	306	2	-	-	PUNCT
ajird-398	306	3	qism	qism	NOUN
ajird-398	306	4	.	.	PUNCT
ajird-398	307	1	“	"	PUNCT
ajird-398	307	2	o`zbekiston	o`zbekiston	PROPN
ajird-398	307	3	”	"	PUNCT
ajird-398	307	4	nashriyoti	nashriyoti	PROPN
ajird-398	307	5	.	.	PUNCT
ajird-398	308	1	toshkent	toshkent	PROPN
ajird-398	308	2	,	,	PUNCT
ajird-398	308	3	1993	1993	NUM
ajird-398	308	4	.	.	PUNCT
ajird-398	309	1	4	4	NUM
ajird-398	309	2	.	.	X
ajird-398	309	3	weisstein	weisstein	PROPN
ajird-398	309	4	,	,	PUNCT
ajird-398	309	5	eric	eric	PROPN
ajird-398	309	6	w.	w.	PROPN
ajird-398	309	7	"	"	PUNCT
ajird-398	309	8	taylor	taylor	PROPN
ajird-398	309	9	series	series	PROPN
ajird-398	309	10	"	"	PUNCT
ajird-398	309	11	.	.	PUNCT
ajird-398	310	1	mathworld	mathworld	NOUN
ajird-398	310	2	.	.	PUNCT
ajird-398	311	1	5	5	X
ajird-398	311	2	.	.	X
ajird-398	311	3	guoning	guone	VERB
ajird-398	311	4	wu	wu	PROPN
ajird-398	311	5	.	.	PUNCT
ajird-398	312	1	integrals	integral	NOUN
ajird-398	312	2	depending	depend	VERB
ajird-398	312	3	on	on	ADP
ajird-398	312	4	a	a	DET
ajird-398	312	5	parameter	parameter	NOUN
ajird-398	312	6	.	.	PUNCT
ajird-398	313	1	china	china	PROPN
ajird-398	313	2	university	university	PROPN
ajird-398	313	3	of	of	ADP
ajird-398	313	4	petroleumbeijing	petroleumbeije	VERB
ajird-398	313	5	2017.9	2017.9	NUM
ajird-398	313	6	.	.	PUNCT
ajird-398	314	1	https://en.wikipedia.org/wiki/eric_w._weisstein	https://en.wikipedia.org/wiki/eric_w._weisstein	PROPN
ajird-398	314	2	https://mathworld.wolfram.com/taylorseries.html	https://mathworld.wolfram.com/taylorseries.html	PROPN
ajird-398	314	3	https://en.wikipedia.org/wiki/mathworld	https://en.wikipedia.org/wiki/mathworld	NOUN