4 Elementary Functions Trigonometric Functions 4.25 Continued Fractions 4.27 Sums

§4.26 Integrals

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Permalink:: http://dlmf.nist.gov/4.26
See also:: Annotations for Ch.4

§4.26(i) Introduction

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Permalink:: http://dlmf.nist.gov/4.26.i
See also:: Annotations for §4.26 and Ch.4

Throughout this section the variables are assumed to be real. The results in §§4.26(ii) and 4.26(iv) can be extended to the complex plane by using continuous branches and avoiding singularities.

§4.26(ii) Indefinite Integrals

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Keywords:: indefinite, integrals, trigonometric functions
Notes:: These results may be verified by differentiation.
Referenced by:: §1.4(iv), §4.26(i)
Permalink:: http://dlmf.nist.gov/4.26.ii
See also:: Annotations for §4.26 and Ch.4

4.26.1	$\displaystyle\int\sin x\,\mathrm{d}x$	$\displaystyle=-\cos x,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function and $x$ : real variable A&S Ref: 4.3.113 Permalink: http://dlmf.nist.gov/4.26.E1 Encodings: TeX, pMML, png See also: Annotations for §4.26(ii), §4.26 and Ch.4
4.26.2	$\displaystyle\int\cos x\,\mathrm{d}x$	$\displaystyle=\sin x.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function and $x$ : real variable A&S Ref: 4.3.114 Permalink: http://dlmf.nist.gov/4.26.E2 Encodings: TeX, pMML, png See also: Annotations for §4.26(ii), §4.26 and Ch.4
4.26.3	$\displaystyle\int\tan x\,\mathrm{d}x$	$\displaystyle=-\ln\left(\cos x\right),$
$-\tfrac{1}{2}\pi<x<\tfrac{1}{2}\pi$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\tan\NVar{z}$ : tangent function and $x$ : real variable A&S Ref: 4.3.115 Permalink: http://dlmf.nist.gov/4.26.E3 Encodings: TeX, pMML, png See also: Annotations for §4.26(ii), §4.26 and Ch.4
4.26.4	$\displaystyle\int\csc x\,\mathrm{d}x$	$\displaystyle=\ln\left(\tan\tfrac{1}{2}x\right),$
$0<x<\pi$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\tan\NVar{z}$ : tangent function and $x$ : real variable A&S Ref: 4.3.116 Permalink: http://dlmf.nist.gov/4.26.E4 Encodings: TeX, pMML, png See also: Annotations for §4.26(ii), §4.26 and Ch.4

4.26.5		$\int\sec x\,\mathrm{d}x={\operatorname{gd}^{-1}}\left(x\right),$
$-\frac{1}{2}\pi<x<\frac{1}{2}\pi$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, ${\operatorname{gd}^{-1}}\left(\NVar{x}\right)$ : inverse Gudermannian function, $\sec\NVar{z}$ : secant function and $x$ : real variable A&S Ref: 4.3.117 Permalink: http://dlmf.nist.gov/4.26.E5 Encodings: TeX, pMML, png See also: Annotations for §4.26(ii), §4.26 and Ch.4

For the right-hand side see (4.23.41) and (4.23.42).

4.26.6		$\int\cot x\,\mathrm{d}x=\ln\left(\sin x\right),$
$0<x<\pi$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function and $x$ : real variable A&S Ref: 4.3.118 Permalink: http://dlmf.nist.gov/4.26.E6 Encodings: TeX, pMML, png See also: Annotations for §4.26(ii), §4.26 and Ch.4

4.26.7		$\int e^{ax}\sin\left(bx\right)\,\mathrm{d}x=\frac{e^{ax}}{a^{2}+b^{2}}(a\sin% \left(bx\right)-b\cos\left(bx\right)),$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\sin\NVar{z}$ : sine function, $a$ : real or complex constant and $x$ : real variable A&S Ref: 4.3.136 Permalink: http://dlmf.nist.gov/4.26.E7 Encodings: TeX, pMML, png See also: Annotations for §4.26(ii), §4.26 and Ch.4

4.26.8		$\int e^{ax}\cos\left(bx\right)\,\mathrm{d}x=\frac{e^{ax}}{a^{2}+b^{2}}(a\cos% \left(bx\right)+b\sin\left(bx\right)).$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\sin\NVar{z}$ : sine function, $a$ : real or complex constant and $x$ : real variable A&S Ref: 4.3.137 Permalink: http://dlmf.nist.gov/4.26.E8 Encodings: TeX, pMML, png See also: Annotations for §4.26(ii), §4.26 and Ch.4

§4.26(iii) Definite Integrals

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Keywords:: definite, integrals, trigonometric functions
Notes:: For (4.26.12) and (4.26.13) see Copson (1935, pp. 137 and 227).
Permalink:: http://dlmf.nist.gov/4.26.iii
See also:: Annotations for §4.26 and Ch.4

Throughout this subsection $m$ and $n$ are integers.

Orthogonality Properties

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Keywords:: orthogonality, trigonometric functions
See also:: Annotations for §4.26(iii), §4.26 and Ch.4

4.26.9		$\int_{0}^{\pi}\sin\left(mt\right)\sin\left(nt\right)\,\mathrm{d}t=0,$
$m\neq n$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $m$ : integer and $n$ : integer A&S Ref: 4.3.140 Permalink: http://dlmf.nist.gov/4.26.E9 Encodings: TeX, pMML, png See also: Annotations for §4.26(iii), §4.26(iii), §4.26 and Ch.4

4.26.10		$\int_{0}^{\pi}\cos\left(mt\right)\cos\left(nt\right)\,\mathrm{d}t=0,$
$m\neq n$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $m$ : integer and $n$ : integer A&S Ref: 4.3.140 Permalink: http://dlmf.nist.gov/4.26.E10 Encodings: TeX, pMML, png See also: Annotations for §4.26(iii), §4.26(iii), §4.26 and Ch.4

4.26.11		$\int_{0}^{\pi}{\sin}^{2}\left(nt\right)\,\mathrm{d}t=\int_{0}^{\pi}{\cos}^{2}% \left(nt\right)\,\mathrm{d}t=\tfrac{1}{2}\pi,$
$n\neq 0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function and $n$ : integer A&S Ref: 4.3.141 Permalink: http://dlmf.nist.gov/4.26.E11 Encodings: TeX, pMML, png See also: Annotations for §4.26(iii), §4.26(iii), §4.26 and Ch.4

4.26.12		$\int_{0}^{\infty}\frac{\sin\left(mt\right)}{t}\,\mathrm{d}t=\begin{cases}\frac% {1}{2}\pi,&m>0,\\ 0,&m=0,\\ -\frac{1}{2}\pi,&m<0.\end{cases}$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function and $m$ : integer A&S Ref: 4.3.142 Referenced by: §4.26(iii) Permalink: http://dlmf.nist.gov/4.26.E12 Encodings: TeX, pMML, png See also: Annotations for §4.26(iii), §4.26(iii), §4.26 and Ch.4

4.26.13		$\int_{0}^{\infty}\sin\left(t^{2}\right)\,\mathrm{d}t=\int_{0}^{\infty}\cos% \left(t^{2}\right)\,\mathrm{d}t=\frac{1}{2}\sqrt{\frac{\pi}{2}}.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\sin\NVar{z}$ : sine function A&S Ref: 4.3.144 Referenced by: §4.26(iii) Permalink: http://dlmf.nist.gov/4.26.E13 Encodings: TeX, pMML, png See also: Annotations for §4.26(iii), §4.26(iii), §4.26 and Ch.4

§4.26(iv) Inverse Trigonometric Functions

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Keywords:: integrals, inverse trigonometric functions
Notes:: These results may be verified by differentiation.
Referenced by:: §1.4(iv), §4.26(i)
Permalink:: http://dlmf.nist.gov/4.26.iv
See also:: Annotations for §4.26 and Ch.4

4.26.14		$\int\operatorname{arcsin}x\,\mathrm{d}x=x\operatorname{arcsin}x+(1-x^{2})^{1/2},$
$-1<x<1$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arcsin}\NVar{z}$ : arcsine function and $x$ : real variable A&S Ref: 4.4.58 (modified) Permalink: http://dlmf.nist.gov/4.26.E14 Encodings: TeX, pMML, png See also: Annotations for §4.26(iv), §4.26 and Ch.4

4.26.15		$\int\operatorname{arccos}x\,\mathrm{d}x=x\operatorname{arccos}x-(1-x^{2})^{1/2},$
$-1<x<1$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arccos}\NVar{z}$ : arccosine function and $x$ : real variable A&S Ref: 4.4.59 (modified) Permalink: http://dlmf.nist.gov/4.26.E15 Encodings: TeX, pMML, png See also: Annotations for §4.26(iv), §4.26 and Ch.4

4.26.16		$\int\operatorname{arctan}x\,\mathrm{d}x=x\operatorname{arctan}x-\tfrac{1}{2}% \ln\left(1+x^{2}\right),$
$-\infty<x<\infty$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable A&S Ref: 4.4.60 (modified) Permalink: http://dlmf.nist.gov/4.26.E16 Encodings: TeX, pMML, png See also: Annotations for §4.26(iv), §4.26 and Ch.4

4.26.17		$\int\operatorname{arccsc}x\,\mathrm{d}x=x\operatorname{arccsc}x+\ln\left(x+(x^% {2}-1)^{1/2}\right),$
$1<x<\infty$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arccsc}\NVar{z}$ : arccosecant function, $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable A&S Ref: 4.4.61 (is stated differently.) Permalink: http://dlmf.nist.gov/4.26.E17 Encodings: TeX, pMML, png See also: Annotations for §4.26(iv), §4.26 and Ch.4

4.26.18		$\int\operatorname{arcsec}x\,\mathrm{d}x=x\operatorname{arcsec}x-\ln\left(x+(x^% {2}-1)^{1/2}\right),$
$1<x<\infty$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arcsec}\NVar{z}$ : arcsecant function, $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable A&S Ref: 4.4.62 (is stated differently.) Permalink: http://dlmf.nist.gov/4.26.E18 Encodings: TeX, pMML, png See also: Annotations for §4.26(iv), §4.26 and Ch.4

4.26.19		$\int\operatorname{arccot}x\,\mathrm{d}x=x\operatorname{arccot}x+\tfrac{1}{2}% \ln\left(1+x^{2}\right),$
$0<x<\infty$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arccot}\NVar{z}$ : arccotangent function, $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable A&S Ref: 4.4.63 (modified) Permalink: http://dlmf.nist.gov/4.26.E19 Encodings: TeX, pMML, png See also: Annotations for §4.26(iv), §4.26 and Ch.4

4.26.20		$\int x\operatorname{arcsin}x\,\mathrm{d}x=\left(\frac{x^{2}}{2}-\frac{1}{4}% \right)\operatorname{arcsin}x+\frac{x}{4}(1-x^{2})^{1/2},$
$-1<x<1$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arcsin}\NVar{z}$ : arcsine function and $x$ : real variable A&S Ref: 4.4.64 (modified) Permalink: http://dlmf.nist.gov/4.26.E20 Encodings: TeX, pMML, png See also: Annotations for §4.26(iv), §4.26 and Ch.4

4.26.21		$\int x\operatorname{arccos}x\,\mathrm{d}x=\left(\frac{x^{2}}{2}-\frac{1}{4}% \right)\operatorname{arccos}x-\frac{x}{4}(1-x^{2})^{1/2},$
$-1<x<1$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{arccos}\NVar{z}$ : arccosine function and $x$ : real variable A&S Ref: 4.4.66 (modified) Permalink: http://dlmf.nist.gov/4.26.E21 Encodings: TeX, pMML, png See also: Annotations for §4.26(iv), §4.26 and Ch.4

§4.26(v) Compendia

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Permalink:: http://dlmf.nist.gov/4.26.v
See also:: Annotations for §4.26 and Ch.4

Extensive compendia of indefinite and definite integrals of trigonometric and inverse trigonometric functions include Apelblat (1983, pp. 48–109), Bierens de Haan (1939), Gradshteyn and Ryzhik (2015, Chapters 2–4), Gröbner and Hofreiter (1949, pp. 116–139), Gröbner and Hofreiter (1950, pp. 94–160), and Prudnikov et al. (1986a, §§1.5, 1.7, 2.5, 2.7).

4.25 Continued Fractions 4.27 Sums

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