List of integrals of trigonometric functions

Trigonometry

Outline History Usage Functions (sin, cos, tan, inverse) Generalized trigonometry
Reference
Identities Exact constants Tables Unit circle
Laws and theorems
Sines Cosines Tangents Cotangents Pythagorean theorem
Calculus
Trigonometric substitution Integrals (inverse functions) Derivatives Trigonometric series
Mathematicians
Hipparchus Ptolemy Brahmagupta al-Hasib al-Battani Regiomontanus Viète de Moivre Euler Fourier
v t e

The following is a list of integrals (antiderivative functions) of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antiderivative functions, see Lists of integrals. For the special antiderivatives involving trigonometric functions, see Trigonometric integral.^[1]

Generally, if the function $\sin x$ is any trigonometric function, and $\cos x$ is its derivative,

$\int a\cos nx\,dx={\frac {a}{n}\sin nx+C$

In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.

Integrands involving only sine

$\int \sin ax\,dx=-{\frac {1}{a}\cos ax+C$
$\int \sin ^{2}{ax}\,dx={\frac {x}{2}-{\frac {1}{4a}\sin 2ax+C={\frac {x}{2}-{\frac {1}{2a}\sin ax\cos ax+C$
$\int \sin ^{3}{ax}\,dx={\frac {\cos 3ax}{12a}-{\frac {3\cos ax}{4a}+C$
$\int x\sin ^{2}{ax}\,dx={\frac {x^{2}{4}-{\frac {x}{4a}\sin 2ax-{\frac {1}{8a^{2}\cos 2ax+C$
$\int x^{2}\sin ^{2}{ax}\,dx={\frac {x^{3}{6}-\left({\frac {x^{2}{4a}-{\frac {1}{8a^{3}\right)\sin 2ax-{\frac {x}{4a^{2}\cos 2ax+C$
$\int x\sin ax\,dx={\frac {\sin ax}{a^{2}-{\frac {x\cos ax}{a}+C$
$\int (\sin b_{1}x)(\sin b_{2}x)\,dx={\frac {\sin((b_{2}-b_{1})x)}{2(b_{2}-b_{1})}-{\frac {\sin((b_{1}+b_{2})x)}{2(b_{1}+b_{2})}+C\qquad {\mbox{(for }|b_{1}|\neq |b_{2}|{\mbox{)$
$\int \sin ^{n}{ax}\,dx=-{\frac {\sin ^{n-1}ax\cos ax}{na}+{\frac {n-1}{n}\int \sin ^{n-2}ax\,dx\qquad {\mbox{(for }n>0{\mbox{)$
$\int {\frac {dx}{\sin ax}=-{\frac {1}{a}\ln {\left|\csc {ax}+\cot {ax}\right|}+C$
$\int {\frac {dx}{\sin ^{n}ax}={\frac {\cos ax}{a(1-n)\sin ^{n-1}ax}+{\frac {n-2}{n-1}\int {\frac {dx}{\sin ^{n-2}ax}\qquad {\mbox{(for }n>1{\mbox{)$
${\begin{aligned}\int x^{n}\sin ax\,dx&=-{\frac {x^{n}{a}\cos ax+{\frac {n}{a}\int x^{n-1}\cos ax\,dx\\&=\sum _{k=0}^{2k\leq n}(-1)^{k+1}{\frac {x^{n-2k}{a^{1+2k}{\frac {n!}{(n-2k)!}\cos ax+\sum _{k=0}^{2k+1\leq n}(-1)^{k}{\frac {x^{n-1-2k}{a^{2+2k}{\frac {n!}{(n-2k-1)!}\sin ax\\&=-\sum _{k=0}^{n}{\frac {x^{n-k}{a^{1+k}{\frac {n!}{(n-k)!}\cos \left(ax+k{\frac {\pi }{2}\right)\qquad {\mbox{(for }n>0{\mbox{)}\end{aligned$
$\int {\frac {\sin ax}{x}\,dx=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(ax)^{2n+1}{(2n+1)\cdot (2n+1)!}+C$
$\int {\frac {\sin ax}{x^{n}\,dx=-{\frac {\sin ax}{(n-1)x^{n-1}+{\frac {a}{n-1}\int {\frac {\cos ax}{x^{n-1}\,dx$
$\int {\sin {\mathrm {(} }{ax}^{2}\mathrm {+} {bx}\mathrm {+} {c}{\mathrm {)} }{dx}\mathrm {=} \left\{\begin{aligned}&{\sqrt {a}{\sqrt {\frac {\mathit {\pi }{2}\cos \left({\frac {b}^{2}\mathrm {-} {4}{ac}{4a}\right){S}\left({\frac {2}{ax}\mathrm {+} {b}{\sqrt {2}{a}{\mathit {\pi }\right)\mathrm {+} {\sqrt {a}{\sqrt {\frac {\mathit {\pi }{2}\sin \left({\frac {b}^{2}\mathrm {-} {4}{ac}{4a}\right){C}\left({\frac {2}{ax}\mathrm {+} {b}{\sqrt {2}{a}{\mathit {\pi }\right)\;{to}\;{b}^{2}\mathrm {-} {4}{ac}\;{\mathrm {>} }\;{0}\\&{\sqrt {a}{\sqrt {\frac {\mathit {\pi }{2}\cos \left({\frac {b}^{2}\mathrm {-} {4}{ac}{4a}\right){S}\left({\frac {2}{ax}\mathrm {+} {b}{\sqrt {2}{a}{\mathit {\pi }\right)\mathrm {-} {\sqrt {a}{\sqrt {\frac {\mathit {\pi }{2}\sin \left({\frac {b}^{2}\mathrm {-} {4}{ac}{4a}\right){C}\left({\frac {2}{ax}\mathrm {+} {b}{\sqrt {2}{a}{\mathit {\pi }\right)\;{to}\;{b}^{2}\mathrm {-} {4}{ac}\;{\mathrm {<} }\;{0}\end{aligned}\right.\;\;{for}\;{a}\diagup \!\!\!\!{\mathrm {=} }{0}{\mathrm {,} }\;{a}{\mathrm {>} }{0$
$\int {\frac {dx}{1\pm \sin ax}={\frac {1}{a}\tan \left({\frac {ax}{2}\mp {\frac {\pi }{4}\right)+C$
$\int {\frac {x\,dx}{1+\sin ax}={\frac {x}{a}\tan \left({\frac {ax}{2}-{\frac {\pi }{4}\right)+{\frac {2}{a^{2}\ln \left|\cos \left({\frac {ax}{2}-{\frac {\pi }{4}\right)\right|+C$
$\int {\frac {x\,dx}{1-\sin ax}={\frac {x}{a}\cot \left({\frac {\pi }{4}-{\frac {ax}{2}\right)+{\frac {2}{a^{2}\ln \left|\sin \left({\frac {\pi }{4}-{\frac {ax}{2}\right)\right|+C$
$\int {\frac {\sin ax\,dx}{1\pm \sin ax}=\pm x+{\frac {1}{a}\tan \left({\frac {\pi }{4}\mp {\frac {ax}{2}\right)+C$

Integrands involving only cosine

$\int \cos ax\,dx={\frac {1}{a}\sin ax+C$
$\int \cos ^{2}{ax}\,dx={\frac {x}{2}+{\frac {1}{4a}\sin 2ax+C={\frac {x}{2}+{\frac {1}{2a}\sin ax\cos ax+C$
$\int \cos ^{n}ax\,dx={\frac {\cos ^{n-1}ax\sin ax}{na}+{\frac {n-1}{n}\int \cos ^{n-2}ax\,dx\qquad {\mbox{(for }n>0{\mbox{)$
$\int x\cos ax\,dx={\frac {\cos ax}{a^{2}+{\frac {x\sin ax}{a}+C$
$\int x^{2}\cos ^{2}{ax}\,dx={\frac {x^{3}{6}+\left({\frac {x^{2}{4a}-{\frac {1}{8a^{3}\right)\sin 2ax+{\frac {x}{4a^{2}\cos 2ax+C$
${\begin{aligned}\int x^{n}\cos ax\,dx&={\frac {x^{n}\sin ax}{a}-{\frac {n}{a}\int x^{n-1}\sin ax\,dx\\&=\sum _{k=0}^{2k+1\leq n}(-1)^{k}{\frac {x^{n-2k-1}{a^{2+2k}{\frac {n!}{(n-2k-1)!}\cos ax+\sum _{k=0}^{2k\leq n}(-1)^{k}{\frac {x^{n-2k}{a^{1+2k}{\frac {n!}{(n-2k)!}\sin ax\\&=\sum _{k=0}^{n}(-1)^{\lfloor k/2\rfloor }{\frac {x^{n-k}{a^{1+k}{\frac {n!}{(n-k)!}\cos \left(ax-{\frac {(-1)^{k}+1}{2}{\frac {\pi }{2}\right)\\&=\sum _{k=0}^{n}{\frac {x^{n-k}{a^{1+k}{\frac {n!}{(n-k)!}\sin \left(ax+k{\frac {\pi }{2}\right)\qquad {\mbox{(for }n>0{\mbox{)}\end{aligned$
$\int {\frac {\cos ax}{x}\,dx=\ln |ax|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(ax)^{2k}{2k\cdot (2k)!}+C$
$\int {\frac {\cos ax}{x^{n}\,dx=-{\frac {\cos ax}{(n-1)x^{n-1}-{\frac {a}{n-1}\int {\frac {\sin ax}{x^{n-1}\,dx\qquad {\mbox{(for }n\neq 1{\mbox{)$
$\int {\frac {dx}{\cos ax}={\frac {1}{a}\ln \left|\tan \left({\frac {ax}{2}+{\frac {\pi }{4}\right)\right|+C$
$\int {\frac {dx}{\cos ^{n}ax}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}+{\frac {n-2}{n-1}\int {\frac {dx}{\cos ^{n-2}ax}\qquad {\mbox{(for }n>1{\mbox{)$
$\int {\frac {dx}{1+\cos ax}={\frac {1}{a}\tan {\frac {ax}{2}+C$
$\int {\frac {dx}{1-\cos ax}=-{\frac {1}{a}\cot {\frac {ax}{2}+C$
$\int {\frac {x\,dx}{1+\cos ax}={\frac {x}{a}\tan {\frac {ax}{2}+{\frac {2}{a^{2}\ln \left|\cos {\frac {ax}{2}\right|+C$
$\int {\frac {x\,dx}{1-\cos ax}=-{\frac {x}{a}\cot {\frac {ax}{2}+{\frac {2}{a^{2}\ln \left|\sin {\frac {ax}{2}\right|+C$
$\int {\frac {\cos ax\,dx}{1+\cos ax}=x-{\frac {1}{a}\tan {\frac {ax}{2}+C$
$\int {\frac {\cos ax\,dx}{1-\cos ax}=-x-{\frac {1}{a}\cot {\frac {ax}{2}+C$
$\int (\cos a_{1}x)(\cos a_{2}x)\,dx={\frac {\sin((a_{2}-a_{1})x)}{2(a_{2}-a_{1})}+{\frac {\sin((a_{2}+a_{1})x)}{2(a_{2}+a_{1})}+C\qquad {\mbox{(for }|a_{1}|\neq |a_{2}|{\mbox{)$

Integrands involving only tangent

$\int \tan ax\,dx=-{\frac {1}{a}\ln |\cos ax|+C={\frac {1}{a}\ln |\sec ax|+C$
$\int \tan ^{2}{x}\,dx=\tan {x}-x+C$
$\int \tan ^{n}ax\,dx={\frac {1}{a(n-1)}\tan ^{n-1}ax-\int \tan ^{n-2}ax\,dx\qquad {\mbox{(for }n\neq 1{\mbox{)$
$\int {\frac {dx}{q\tan ax+p}={\frac {1}{p^{2}+q^{2}(px+{\frac {q}{a}\ln |q\sin ax+p\cos ax|)+C\qquad {\mbox{(for }p^{2}+q^{2}\neq 0{\mbox{)$
$\int {\frac {dx}{\tan ax\pm 1}=\pm {\frac {x}{2}+{\frac {1}{2a}\ln |\sin ax\pm \cos ax|+C$
$\int {\frac {\tan ax\,dx}{\tan ax\pm 1}={\frac {x}{2}\mp {\frac {1}{2a}\ln |\sin ax\pm \cos ax|+C$

Integrands involving only secant

$\int \sec {ax}\,dx={\frac {1}{a}\ln {\left|\sec {ax}+\tan {ax}\right|}+C={\frac {1}{a}\ln {\left|\tan {\left({\frac {ax}{2}+{\frac {\pi }{4}\right)}\right|}+C={\frac {1}{a}\operatorname {artanh} {\left(\sin {ax}\right)}+C$
$\int \sec ^{2}{x}\,dx=\tan {x}+C$
$\int \sec ^{3}{x}\,dx={\frac {1}{2}\sec x\tan x+{\frac {1}{2}\ln |\sec x+\tan x|+C.$
$\int \sec ^{n}{ax}\,dx={\frac {\sec ^{n-2}{ax}\tan {ax}{a(n-1)}\,+\,{\frac {n-2}{n-1}\int \sec ^{n-2}{ax}\,dx\qquad {\mbox{ (for }n\neq 1{\mbox{)$
$\int {\frac {dx}{\sec {x}+1}=x-\tan {\frac {x}{2}+C$
$\int {\frac {dx}{\sec {x}-1}=-x-\cot {\frac {x}{2}+C$

$\int {\frac {\sin {x}{\cos {x}=\int \tan {x$

Integrands involving only cosecant

$\int \csc {ax}\,dx=-{\frac {1}{a}\ln {\left|\csc {ax}+\cot {ax}\right|}+C={\frac {1}{a}\ln {\left|\csc {ax}-\cot {ax}\right|}+C={\frac {1}{a}\ln {\left|\tan {\left({\frac {ax}{2}\right)}\right|}+C$
$\int \csc ^{2}{x}\,dx=-\cot {x}+C$
$\int \csc ^{3}{x}\,dx=-{\frac {1}{2}\csc x\cot x-{\frac {1}{2}\ln |\csc x+\cot x|+C=-{\frac {1}{2}\csc x\cot x+{\frac {1}{2}\ln |\csc x-\cot x|+C$
$\int \csc ^{n}{ax}\,dx=-{\frac {\csc ^{n-2}{ax}\cot {ax}{a(n-1)}\,+\,{\frac {n-2}{n-1}\int \csc ^{n-2}{ax}\,dx\qquad {\mbox{ (for }n\neq 1{\mbox{)$
$\int {\frac {dx}{\csc {x}+1}=x-{\frac {2}{\cot {\frac {x}{2}+1}+C$
$\int {\frac {dx}{\csc {x}-1}=-x+{\frac {2}{\cot {\frac {x}{2}-1}+C$

Integrands involving only cotangent

$\int \cot ax\,dx={\frac {1}{a}\ln |\sin ax|+C$
$\int \cot ^{2}{x}\,dx=-\cot {x}-x+C$
$\int \cot ^{n}ax\,dx=-{\frac {1}{a(n-1)}\cot ^{n-1}ax-\int \cot ^{n-2}ax\,dx\qquad {\mbox{(for }n\neq 1{\mbox{)$
$\int {\frac {dx}{1+\cot ax}=\int {\frac {\tan ax\,dx}{\tan ax+1}={\frac {x}{2}-{\frac {1}{2a}\ln |\sin ax+\cos ax|+C$
$\int {\frac {dx}{1-\cot ax}=\int {\frac {\tan ax\,dx}{\tan ax-1}={\frac {x}{2}+{\frac {1}{2a}\ln |\sin ax-\cos ax|+C$

Integrands involving both sine and cosine

An integral that is a rational function of the sine and cosine can be evaluated using Bioche's rules.

$\int {\frac {dx}{\cos ax\pm \sin ax}={\frac {1}{a{\sqrt {2}\ln \left|\tan \left({\frac {ax}{2}\pm {\frac {\pi }{8}\right)\right|+C$
$\int {\frac {dx}{(\cos ax\pm \sin ax)^{2}={\frac {1}{2a}\tan \left(ax\mp {\frac {\pi }{4}\right)+C$
$\int {\frac {dx}{(\cos x+\sin x)^{n}={\frac {1}{2(n-1)}\left({\frac {\sin x-\cos x}{(\cos x+\sin x)^{n-1}+(n-2)\int {\frac {dx}{(\cos x+\sin x)^{n-2}\right)$
$\int {\frac {\cos ax\,dx}{\cos ax+\sin ax}={\frac {x}{2}+{\frac {1}{2a}\ln \left|\sin ax+\cos ax\right|+C$
$\int {\frac {\cos ax\,dx}{\cos ax-\sin ax}={\frac {x}{2}-{\frac {1}{2a}\ln \left|\sin ax-\cos ax\right|+C$
$\int {\frac {\sin ax\,dx}{\cos ax+\sin ax}={\frac {x}{2}-{\frac {1}{2a}\ln \left|\sin ax+\cos ax\right|+C$
$\int {\frac {\sin ax\,dx}{\cos ax-\sin ax}=-{\frac {x}{2}-{\frac {1}{2a}\ln \left|\sin ax-\cos ax\right|+C$
$\int {\frac {\cos ax\,dx}{(\sin ax)(1+\cos ax)}=-{\frac {1}{4a}\tan ^{2}{\frac {ax}{2}+{\frac {1}{2a}\ln \left|\tan {\frac {ax}{2}\right|+C$
$\int {\frac {\cos ax\,dx}{(\sin ax)(1-\cos ax)}=-{\frac {1}{4a}\cot ^{2}{\frac {ax}{2}-{\frac {1}{2a}\ln \left|\tan {\frac {ax}{2}\right|+C$
$\int {\frac {\sin ax\,dx}{(\cos ax)(1+\sin ax)}={\frac {1}{4a}\cot ^{2}\left({\frac {ax}{2}+{\frac {\pi }{4}\right)+{\frac {1}{2a}\ln \left|\tan \left({\frac {ax}{2}+{\frac {\pi }{4}\right)\right|+C$
$\int {\frac {\sin ax\,dx}{(\cos ax)(1-\sin ax)}={\frac {1}{4a}\tan ^{2}\left({\frac {ax}{2}+{\frac {\pi }{4}\right)-{\frac {1}{2a}\ln \left|\tan \left({\frac {ax}{2}+{\frac {\pi }{4}\right)\right|+C$
$\int (\sin ax)(\cos ax)\,dx={\frac {1}{2a}\sin ^{2}ax+C$
$\int (\sin a_{1}x)(\cos a_{2}x)\,dx=-{\frac {\cos((a_{1}-a_{2})x)}{2(a_{1}-a_{2})}-{\frac {\cos((a_{1}+a_{2})x)}{2(a_{1}+a_{2})}+C\qquad {\mbox{(for }|a_{1}|\neq |a_{2}|{\mbox{)$
$\int (\sin ^{n}ax)(\cos ax)\,dx={\frac {1}{a(n+1)}\sin ^{n+1}ax+C\qquad {\mbox{(for }n\neq -1{\mbox{)$
$\int (\sin ax)(\cos ^{n}ax)\,dx=-{\frac {1}{a(n+1)}\cos ^{n+1}ax+C\qquad {\mbox{(for }n\neq -1{\mbox{)$
${\begin{aligned}\int (\sin ^{n}ax)(\cos ^{m}ax)\,dx&=-{\frac {(\sin ^{n-1}ax)(\cos ^{m+1}ax)}{a(n+m)}+{\frac {n-1}{n+m}\int (\sin ^{n-2}ax)(\cos ^{m}ax)\,dx\qquad {\mbox{(for }m,n>0{\mbox{)}\\&={\frac {(\sin ^{n+1}ax)(\cos ^{m-1}ax)}{a(n+m)}+{\frac {m-1}{n+m}\int (\sin ^{n}ax)(\cos ^{m-2}ax)\,dx\qquad {\mbox{(for }m,n>0{\mbox{)}\end{aligned$
$\int {\frac {dx}{(\sin ax)(\cos ax)}={\frac {1}{a}\ln \left|\tan ax\right|+C$
$\int {\frac {dx}{(\sin ax)(\cos ^{n}ax)}={\frac {1}{a(n-1)\cos ^{n-1}ax}+\int {\frac {dx}{(\sin ax)(\cos ^{n-2}ax)}\qquad {\mbox{(for }n\neq 1{\mbox{)$
$\int {\frac {dx}{(\sin ^{n}ax)(\cos ax)}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}+\int {\frac {dx}{(\sin ^{n-2}ax)(\cos ax)}\qquad {\mbox{(for }n\neq 1{\mbox{)$
$\int {\frac {\sin ax\,dx}{\cos ^{n}ax}={\frac {1}{a(n-1)\cos ^{n-1}ax}+C\qquad {\mbox{(for }n\neq 1{\mbox{)$
$\int {\frac {\sin ^{2}ax\,dx}{\cos ax}=-{\frac {1}{a}\sin ax+{\frac {1}{a}\ln \left|\tan \left({\frac {\pi }{4}+{\frac {ax}{2}\right)\right|+C$
$\int {\frac {\sin ^{2}ax\,dx}{\cos ^{n}ax}={\frac {\sin ax}{a(n-1)\cos ^{n-1}ax}-{\frac {1}{n-1}\int {\frac {dx}{\cos ^{n-2}ax}\qquad {\mbox{(for }n\neq 1{\mbox{)$
${\begin{aligned}\int {\frac {\sin ^{2}x}{1+\cos ^{2}x}\,dx&={\sqrt {2}\operatorname {arctangant} \left({\frac {\tan x}{\sqrt {2}\right)-x\qquad {\mbox{(for x in}]-{\frac {\pi }{2};+{\frac {\pi }{2}[{\mbox{)}\\&={\sqrt {2}\operatorname {arctangant} \left({\frac {\tan x}{\sqrt {2}\right)-\operatorname {arctangant} \left(\tan x\right)\qquad {\mbox{(this time x being any real number }{\mbox{)}\end{aligned$
$\int {\frac {\sin ^{n}ax\,dx}{\cos ax}=-{\frac {\sin ^{n-1}ax}{a(n-1)}+\int {\frac {\sin ^{n-2}ax\,dx}{\cos ax}\qquad {\mbox{(for }n\neq 1{\mbox{)$
$\int {\frac {\sin ^{n}ax\,dx}{\cos ^{m}ax}={\begin{cases}{\frac {\sin ^{n+1}ax}{a(m-1)\cos ^{m-1}ax}-{\frac {n-m+2}{m-1}\int {\frac {\sin ^{n}ax\,dx}{\cos ^{m-2}ax}&{\mbox{(for }m\neq 1{\mbox{)}\\{\frac {\sin ^{n-1}ax}{a(m-1)\cos ^{m-1}ax}-{\frac {n-1}{m-1}\int {\frac {\sin ^{n-2}ax\,dx}{\cos ^{m-2}ax}&{\mbox{(for }m\neq 1{\mbox{)}\\-{\frac {\sin ^{n-1}ax}{a(n-m)\cos ^{m-1}ax}+{\frac {n-1}{n-m}\int {\frac {\sin ^{n-2}ax\,dx}{\cos ^{m}ax}&{\mbox{(for }m\neq n{\mbox{)}\end{cases$
$\int {\frac {\cos ax\,dx}{\sin ^{n}ax}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}+C\qquad {\mbox{(for }n\neq 1{\mbox{)$
$\int {\frac {\cos ^{2}ax\,dx}{\sin ax}={\frac {1}{a}\left(\cos ax+\ln \left|\tan {\frac {ax}{2}\right|\right)+C$
$\int {\frac {\cos ^{2}ax\,dx}{\sin ^{n}ax}=-{\frac {1}{n-1}\left({\frac {\cos ax}{a\sin ^{n-1}ax}+\int {\frac {dx}{\sin ^{n-2}ax}\right)\qquad {\mbox{(for }n\neq 1{\mbox{)$
$\int {\frac {\cos ^{n}ax\,dx}{\sin ^{m}ax}={\begin{cases}-{\frac {\cos ^{n+1}ax}{a(m-1)\sin ^{m-1}ax}-{\frac {n-m+2}{m-1}\int {\frac {\cos ^{n}ax\,dx}{\sin ^{m-2}ax}&{\mbox{(for }n\neq 1{\mbox{)}\\-{\frac {\cos ^{n-1}ax}{a(m-1)\sin ^{m-1}ax}-{\frac {n-1}{m-1}\int {\frac {\cos ^{n-2}ax\,dx}{\sin ^{m-2}ax}&{\mbox{(for }m\neq 1{\mbox{)}\\{\frac {\cos ^{n-1}ax}{a(n-m)\sin ^{m-1}ax}+{\frac {n-1}{n-m}\int {\frac {\cos ^{n-2}ax\,dx}{\sin ^{m}ax}&{\mbox{(for }m\neq n{\mbox{)}\end{cases$

Integrands involving both sine and tangent

$\int (\sin ax)(\tan ax)\,dx={\frac {1}{a}(\ln |\sec ax+\tan ax|-\sin ax)+C$
$\int {\frac {\tan ^{n}ax\,dx}{\sin ^{2}ax}={\frac {1}{a(n-1)}\tan ^{n-1}(ax)+C\qquad {\mbox{(for }n\neq 1{\mbox{)$

Integrand involving both cosine and tangent

$\int {\frac {\tan ^{n}ax\,dx}{\cos ^{2}ax}={\frac {1}{a(n+1)}\tan ^{n+1}ax+C\qquad {\mbox{(for }n\neq -1{\mbox{)$

Integrand involving both sine and cotangent

$\int {\frac {\cot ^{n}ax\,dx}{\sin ^{2}ax}=-{\frac {1}{a(n+1)}\cot ^{n+1}ax+C\qquad {\mbox{(for }n\neq -1{\mbox{)$

Integrand involving both cosine and cotangent

$\int {\frac {\cot ^{n}ax\,dx}{\cos ^{2}ax}={\frac {1}{a(1-n)}\tan ^{1-n}ax+C\qquad {\mbox{(for }n\neq 1{\mbox{)$

Integrand involving both secant and tangent

$\int (\sec x)(\tan x)\,dx=\sec x+C$

Integrand involving both cosecant and cotangent

$\int (\csc x)(\cot x)\,dx=-\csc x+C$

Integrals in a quarter period

Using the beta function $B(a,b)$ one can write

$\int _{0}^{\frac {\pi }{2}\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2}\cos ^{n}x\,dx={\frac {1}{2}B\left({\frac {n+1}{2},{\frac {1}{2}\right)={\begin{cases}{\frac {n-1}{n}\cdot {\frac {n-3}{n-2}\cdots {\frac {3}{4}\cdot {\frac {1}{2}\cdot {\frac {\pi }{2},&{\text{if }n{\text{ is even}\\{\frac {n-1}{n}\cdot {\frac {n-3}{n-2}\cdots {\frac {4}{5}\cdot {\frac {2}{3},&{\text{if }n{\text{ is odd and more than 1}\\1,&{\text{if }n=1\end{cases$

Integrals with symmetric limits

$\int _{-c}^{c}\sin {x}\,dx=0$
$\int _{-c}^{c}\cos {x}\,dx=2\int _{0}^{c}\cos {x}\,dx=2\int _{-c}^{0}\cos {x}\,dx=2\sin {c$
$\int _{-c}^{c}\tan {x}\,dx=0$
$\int _{-{\frac {a}{2}^{\frac {a}{2}x^{2}\cos ^{2}{\frac {n\pi x}{a}\,dx={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}\qquad {\mbox{(for }n=1,3,5...{\mbox{)$
$\int _{\frac {-a}{2}^{\frac {a}{2}x^{2}\sin ^{2}{\frac {n\pi x}{a}\,dx={\frac {a^{3}(n^{2}\pi ^{2}-6(-1)^{n})}{24n^{2}\pi ^{2}={\frac {a^{3}{24}(1-6{\frac {(-1)^{n}{n^{2}\pi ^{2})\qquad {\mbox{(for }n=1,2,3,...{\mbox{)$

Integral over a full circle

$\int _{0}^{2\pi }\sin ^{2m+1}{x}\cos ^{n}{x}\,dx=0\!\qquad n,m\in \mathbb {Z}$
$\int _{0}^{2\pi }\sin ^{m}{x}\cos ^{2n+1}{x}\,dx=0\!\qquad n,m\in \mathbb {Z}$

See also

Trigonometric integral

References

^ Bresock, Krista (2022-01-01). "Student Understanding of the Definite Integral When Solving Calculus Volume Problems". Graduate Theses, Dissertations, and Problem Reports. doi:10.33915/etd.11491.

v t e Lists of integrals
Rational functions Irrational functions Trigonometric functions Inverse trigonometric functions Hyperbolic functions Inverse hyperbolic functions Exponential functions Logarithmic functions Gaussian functions Definite integrals