# §4.26 Integrals

## §4.26(i) Introduction

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

 4.26.1 $\displaystyle\int\mathop{\sin\/}\nolimits x\mathrm{d}x$ $\displaystyle=-\mathop{\cos\/}\nolimits x,$ 4.26.2 $\displaystyle\int\mathop{\cos\/}\nolimits x\mathrm{d}x$ $\displaystyle=\mathop{\sin\/}\nolimits x.$ 4.26.3 $\displaystyle\int\mathop{\tan\/}\nolimits x\mathrm{d}x$ $\displaystyle=-\mathop{\ln\/}\nolimits\!\left(\mathop{\cos\/}\nolimits x\right),$ $-\tfrac{1}{2}\pi. 4.26.4 $\displaystyle\int\mathop{\csc\/}\nolimits x\mathrm{d}x$ $\displaystyle=\mathop{\ln\/}\nolimits\!\left(\mathop{\tan\/}\nolimits\tfrac{1}% {2}x\right),$ $0.
 4.26.5 $\int\mathop{\sec\/}\nolimits x\mathrm{d}x=\mathop{{\mathrm{gd}^{-1}}\/}% \nolimits\!\left(x\right),$ $-\frac{1}{2}\pi.

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

 4.26.6 $\int\mathop{\cot\/}\nolimits x\mathrm{d}x=\mathop{\ln\/}\nolimits\!\left(% \mathop{\sin\/}\nolimits x\right),$ $0.
 4.26.7 $\int e^{ax}\mathop{\sin\/}\nolimits\!\left(bx\right)\mathrm{d}x=\frac{e^{ax}}{% a^{2}+b^{2}}(a\mathop{\sin\/}\nolimits\!\left(bx\right)-b\mathop{\cos\/}% \nolimits\!\left(bx\right)),$
 4.26.8 $\int e^{ax}\mathop{\cos\/}\nolimits\!\left(bx\right)\mathrm{d}x=\frac{e^{ax}}{% a^{2}+b^{2}}(a\mathop{\cos\/}\nolimits\!\left(bx\right)+b\mathop{\sin\/}% \nolimits\!\left(bx\right)).$

## §4.26(iii) Definite Integrals

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

### Orthogonality Properties

 4.26.9 $\int_{0}^{\pi}\mathop{\sin\/}\nolimits\!\left(mt\right)\mathop{\sin\/}% \nolimits\!\left(nt\right)\mathrm{d}t=0,$ $m\neq n$,
 4.26.10 $\int_{0}^{\pi}\mathop{\cos\/}\nolimits\!\left(mt\right)\mathop{\cos\/}% \nolimits\!\left(nt\right)\mathrm{d}t=0,$ $m\neq n$,
 4.26.11 $\int_{0}^{\pi}{\mathop{\sin\/}\nolimits^{2}}\!\left(nt\right)\mathrm{d}t=\int_% {0}^{\pi}{\mathop{\cos\/}\nolimits^{2}}\!\left(nt\right)\mathrm{d}t=\tfrac{1}{% 2}\pi,$ $n\neq 0$.
 4.26.12 $\int_{0}^{\infty}\frac{\mathop{\sin\/}\nolimits\!\left(mt\right)}{t}\mathrm{d}% t=\begin{cases}\frac{1}{2}\pi,&m>0,\\ 0,&m=0,\\ -\frac{1}{2}\pi,&m<0.\end{cases}$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\sin\/}\nolimits\NVar{z}$: 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)
 4.26.13 $\int_{0}^{\infty}\mathop{\sin\/}\nolimits\!\left(t^{2}\right)\mathrm{d}t=\int_% {0}^{\infty}\mathop{\cos\/}\nolimits\!\left(t^{2}\right)\mathrm{d}t=\frac{1}{2% }\sqrt{\frac{\pi}{2}}.$

## §4.26(iv) Inverse Trigonometric Functions

 4.26.14 $\int\mathop{\mathrm{arcsin}\/}\nolimits x\mathrm{d}x=x\mathop{\mathrm{arcsin}% \/}\nolimits x+(1-x^{2})^{1/2},$ $-1, Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\mathrm{arcsin}\/}\nolimits\NVar{z}$: 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)
 4.26.15 $\int\mathop{\mathrm{arccos}\/}\nolimits x\mathrm{d}x=x\mathop{\mathrm{arccos}% \/}\nolimits x-(1-x^{2})^{1/2},$ $-1. Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\mathrm{arccos}\/}\nolimits\NVar{z}$: 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)
 4.26.16 $\int\mathop{\mathrm{arctan}\/}\nolimits x\mathrm{d}x=x\mathop{\mathrm{arctan}% \/}\nolimits x-\tfrac{1}{2}\mathop{\ln\/}\nolimits\!\left(1+x^{2}\right),$ $-\infty,
 4.26.17 $\int\mathop{\mathrm{arccsc}\/}\nolimits x\mathrm{d}x=x\mathop{\mathrm{arccsc}% \/}\nolimits x+\mathop{\ln\/}\nolimits\!\left(x+(x^{2}-1)^{1/2}\right),$ $1, Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\mathrm{arccsc}\/}\nolimits\NVar{z}$: arccosecant function, $\mathop{\ln\/}\nolimits\NVar{z}$: 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)
 4.26.18 $\int\mathop{\mathrm{arcsec}\/}\nolimits x\mathrm{d}x=x\mathop{\mathrm{arcsec}% \/}\nolimits x-\mathop{\ln\/}\nolimits\!\left(x+(x^{2}-1)^{1/2}\right),$ $1, Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\mathrm{arcsec}\/}\nolimits\NVar{z}$: arcsecant function, $\mathop{\ln\/}\nolimits\NVar{z}$: 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)
 4.26.19 $\int\mathop{\mathrm{arccot}\/}\nolimits x\mathrm{d}x=x\mathop{\mathrm{arccot}% \/}\nolimits x+\tfrac{1}{2}\mathop{\ln\/}\nolimits\!\left(1+x^{2}\right),$ $0.
 4.26.20 $\int x\mathop{\mathrm{arcsin}\/}\nolimits x\mathrm{d}x=\left(\frac{x^{2}}{2}-% \frac{1}{4}\right)\mathop{\mathrm{arcsin}\/}\nolimits x+\frac{x}{4}(1-x^{2})^{% 1/2},$ $-1, Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\mathrm{arcsin}\/}\nolimits\NVar{z}$: 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)
 4.26.21 $\int x\mathop{\mathrm{arccos}\/}\nolimits x\mathrm{d}x=\left(\frac{x^{2}}{2}-% \frac{1}{4}\right)\mathop{\mathrm{arccos}\/}\nolimits x-\frac{x}{4}(1-x^{2})^{% 1/2},$ $-1. Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\mathrm{arccos}\/}\nolimits\NVar{z}$: 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(v) Compendia

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 (2000, 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).