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

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

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

## §4.26(iii) Definite Integrals

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

### Orthogonality Properties

 4.26.9 $\int_{0}^{\pi}\mathop{\sin\/}\nolimits\!\left(mt\right)\mathop{\sin\/}% \nolimits\!\left(nt\right)dt=0,$ $m\neq n$,
 4.26.10 $\int_{0}^{\pi}\mathop{\cos\/}\nolimits\!\left(mt\right)\mathop{\cos\/}% \nolimits\!\left(nt\right)dt=0,$ $m\neq n$,
 4.26.11 $\int_{0}^{\pi}{\mathop{\sin\/}\nolimits^{2}}\!\left(nt\right)dt=\int_{0}^{\pi}% {\mathop{\cos\/}\nolimits^{2}}\!\left(nt\right)dt=\tfrac{1}{2}\pi,$ $n\neq 0$.
 4.26.12 $\int_{0}^{\infty}\frac{\mathop{\sin\/}\nolimits\!\left(mt\right)}{t}dt=\begin{% cases}\frac{1}{2}\pi,&m>0,\\ 0,&m=0,\\ -\frac{1}{2}\pi,&m<0.\end{cases}$ Symbols: $d\NVar{x}$: 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: info for 4.26(iii)
 4.26.13 $\int_{0}^{\infty}\mathop{\sin\/}\nolimits\!\left(t^{2}\right)dt=\int_{0}^{% \infty}\mathop{\cos\/}\nolimits\!\left(t^{2}\right)dt=\frac{1}{2}\sqrt{\frac{% \pi}{2}}.$ Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $d\NVar{x}$: differential of $x$, $\int$: integral and $\mathop{\sin\/}\nolimits\NVar{z}$: 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: info for 4.26(iii)

## §4.26(iv) Inverse Trigonometric Functions

 4.26.14 $\int\mathop{\mathrm{arcsin}\/}\nolimits xdx=x\mathop{\mathrm{arcsin}\/}% \nolimits x+(1-x^{2})^{1/2},$ $-1, Symbols: $d\NVar{x}$: 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: info for 4.26(iv)
 4.26.15 $\int\mathop{\mathrm{arccos}\/}\nolimits xdx=x\mathop{\mathrm{arccos}\/}% \nolimits x-(1-x^{2})^{1/2},$ $-1. Symbols: $d\NVar{x}$: 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: info for 4.26(iv)
 4.26.16 $\int\mathop{\mathrm{arctan}\/}\nolimits xdx=x\mathop{\mathrm{arctan}\/}% \nolimits x-\tfrac{1}{2}\mathop{\ln\/}\nolimits\!\left(1+x^{2}\right),$ $-\infty,
 4.26.17 $\int\mathop{\mathrm{arccsc}\/}\nolimits xdx=x\mathop{\mathrm{arccsc}\/}% \nolimits x+\mathop{\ln\/}\nolimits\!\left(x+(x^{2}-1)^{1/2}\right),$ $1,
 4.26.18 $\int\mathop{\mathrm{arcsec}\/}\nolimits xdx=x\mathop{\mathrm{arcsec}\/}% \nolimits x-\mathop{\ln\/}\nolimits\!\left(x+(x^{2}-1)^{1/2}\right),$ $1,
 4.26.19 $\int\mathop{\mathrm{arccot}\/}\nolimits xdx=x\mathop{\mathrm{arccot}\/}% \nolimits x+\tfrac{1}{2}\mathop{\ln\/}\nolimits\!\left(1+x^{2}\right),$ $0.
 4.26.20 $\int x\mathop{\mathrm{arcsin}\/}\nolimits xdx=\left(\frac{x^{2}}{2}-\frac{1}{4% }\right)\mathop{\mathrm{arcsin}\/}\nolimits x+\frac{x}{4}(1-x^{2})^{1/2},$ $-1, Symbols: $d\NVar{x}$: 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: info for 4.26(iv)
 4.26.21 $\int x\mathop{\mathrm{arccos}\/}\nolimits xdx=\left(\frac{x^{2}}{2}-\frac{1}{4% }\right)\mathop{\mathrm{arccos}\/}\nolimits x-\frac{x}{4}(1-x^{2})^{1/2},$ $-1. Symbols: $d\NVar{x}$: 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: info 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).