List of integrals of trigonometric functions

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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.

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

\int a\cos nx\;\mathrm{d}x = \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[edit]

\int\sin ax\;\mathrm{d}x = -\frac{1}{a}\cos ax+C\,\!


\int\sin^2 {ax}\;\mathrm{d}x = \frac{x}{2} - \frac{1}{4a} \sin 2ax +C= \frac{x}{2} - \frac{1}{2a} \sin ax\cos ax +C\!


\int\sin^3 {ax}\;\mathrm{d}x = \frac{\cos 3ax}{12a} - \frac{3 \cos ax}{4a} +C\!
\int x\sin^2 {ax}\;\mathrm{d}x = \frac{x^2}{4} - \frac{x}{4a} \sin 2ax - \frac{1}{8a^2} \cos 2ax +C\!


\int x^2\sin^2 {ax}\;\mathrm{d}x = \frac{x^3}{6} - \left( \frac {x^2}{4a} - \frac{1}{8a^3} \right) \sin 2ax - \frac{x}{4a^2} \cos 2ax +C\!


\int\sin b_1x\sin b_2x\;\mathrm{d}x = \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}\;\mathrm{d}x = -\frac{\sin^{n-1} ax\cos ax}{na} + \frac{n-1}{n}\int\sin^{n-2} ax\;\mathrm{d}x \qquad\mbox{(for }n>0\mbox{)}\,\!


\int\frac{\mathrm{d}x}{\sin ax} = \frac{1}{a}\ln \left|\tan\frac{ax}{2}\right|+C


\int\frac{\mathrm{d}x}{\sin^n ax} = \frac{\cos ax}{a(1-n) \sin^{n-1} ax}+\frac{n-2}{n-1}\int\frac{\mathrm{d}x}{\sin^{n-2}ax} \qquad\mbox{(for }n>1\mbox{)}\,\!


\int x\sin ax\;\mathrm{d}x = \frac{\sin ax}{a^2}-\frac{x\cos ax}{a}+C\,\!


\int x^n\sin ax\;\mathrm{d}x = -\frac{x^n}{a}\cos ax+\frac{n}{a}\int x^{n-1}\cos ax\;\mathrm{d}x = \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  \qquad\mbox{(for }n>0\mbox{)}\,\!



\int\frac{\sin ax}{x} \mathrm{d}x = \sum_{n=0}^\infty (-1)^n\frac{(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!} +C\,\!


\int\frac{\sin ax}{x^n} \mathrm{d}x = -\frac{\sin ax}{(n-1)x^{n-1}} + \frac{a}{n-1}\int\frac{\cos ax}{x^{n-1}} \mathrm{d}x\,\!


\int\frac{\mathrm{d}x}{1\pm\sin ax} = \frac{1}{a}\tan\left(\frac{ax}{2}\mp\frac{\pi}{4}\right)+C


\int\frac{x\;\mathrm{d}x}{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\;\mathrm{d}x}{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\;\mathrm{d}x}{1\pm\sin ax} = \pm x+\frac{1}{a}\tan\left(\frac{\pi}{4}\mp\frac{ax}{2}\right)+C

Integrands involving only cosine[edit]

\int\cos ax\;\mathrm{d}x = \frac{1}{a}\sin ax+C\,\!
\int\cos^2 {ax}\;\mathrm{d}x = \frac{x}{2} + \frac{1}{4a} \sin 2ax +C = \frac{x}{2} + \frac{1}{2a} \sin ax\cos ax +C\!
\int\cos^n ax\;\mathrm{d}x = \frac{\cos^{n-1} ax\sin ax}{na} + \frac{n-1}{n}\int\cos^{n-2} ax\;\mathrm{d}x \qquad\mbox{(for }n>0\mbox{)}\,\!
\int x\cos ax\;\mathrm{d}x = \frac{\cos ax}{a^2} + \frac{x\sin ax}{a}+C\,\!
\int x^2\cos^2 {ax}\;\mathrm{d}x = \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^n\cos ax\;\mathrm{d}x = \frac{x^n\sin ax}{a} - \frac{n}{a}\int x^{n-1}\sin ax\;\mathrm{d}x\,= \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  \!
\int\frac{\cos ax}{x} \mathrm{d}x = \ln|ax|+\sum_{k=1}^\infty (-1)^k\frac{(ax)^{2k}}{2k\cdot(2k)!}+C\,\!
\int\frac{\cos ax}{x^n} \mathrm{d}x = -\frac{\cos ax}{(n-1)x^{n-1}}-\frac{a}{n-1}\int\frac{\sin ax}{x^{n-1}} \mathrm{d}x \qquad\mbox{(for }n\neq 1\mbox{)}\,\!
\int\frac{\mathrm{d}x}{\cos ax} = \frac{1}{a}\ln\left|\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right)\right|+C
\int\frac{\mathrm{d}x}{\cos^n ax} = \frac{\sin ax}{a(n-1) \cos^{n-1} ax} + \frac{n-2}{n-1}\int\frac{\mathrm{d}x}{\cos^{n-2} ax} \qquad\mbox{(for }n>1\mbox{)}\,\!
\int\frac{\mathrm{d}x}{1+\cos ax} = \frac{1}{a}\tan\frac{ax}{2}+C\,\!
\int\frac{\mathrm{d}x}{1-\cos ax} = -\frac{1}{a}\cot\frac{ax}{2}+C
\int\frac{x\;\mathrm{d}x}{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\;\mathrm{d}x}{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\;\mathrm{d}x}{1+\cos ax} = x - \frac{1}{a}\tan\frac{ax}{2}+C\,\!
\int\frac{\cos ax\;\mathrm{d}x}{1-\cos ax} = -x-\frac{1}{a}\cot\frac{ax}{2}+C\,\!
\int\cos a_1x\cos a_2x\;\mathrm{d}x = \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[edit]

\int\tan ax\;\mathrm{d}x = -\frac{1}{a}\ln|\cos ax|+C = \frac{1}{a}\ln|\sec ax|+C\,\!
\int \tan^2{x} \, \mathrm{d}x = \tan{x} - x +C
\int\tan^n ax\;\mathrm{d}x = \frac{1}{a(n-1)}\tan^{n-1} ax-\int\tan^{n-2} ax\;\mathrm{d}x \qquad\mbox{(for }n\neq 1\mbox{)}\,\!
\int\frac{\mathrm{d}x}{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{\mathrm{d}x}{\tan ax + 1} = \frac{x}{2} + \frac{1}{2a}\ln|\sin ax + \cos ax|+C\,\!
\int\frac{\mathrm{d}x}{\tan ax - 1} = -\frac{x}{2} + \frac{1}{2a}\ln|\sin ax - \cos ax|+C\,\!
\int\frac{\tan ax\;\mathrm{d}x}{\tan ax + 1} = \frac{x}{2} - \frac{1}{2a}\ln|\sin ax + \cos ax|+C\,\!
\int\frac{\tan ax\;\mathrm{d}x}{\tan ax - 1} = \frac{x}{2} + \frac{1}{2a}\ln|\sin ax - \cos ax|+C\,\!

Integrands involving only secant[edit]

See Integral of the secant function.
\int \sec{ax} \, \mathrm{d}x = \frac{1}{a}\ln{\left| \sec{ax} + \tan{ax}\right|}+C
\int \sec^2{x} \, \mathrm{d}x = \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} \, \mathrm{d}x = \frac{\sec^{n-2}{ax} \tan {ax}}{a(n-1)} \,+\, \frac{n-2}{n-1}\int \sec^{n-2}{ax} \, \mathrm{d}x \qquad \mbox{ (for }n \ne 1\mbox{)}\,\!
\int \frac{\mathrm{d}x}{\sec{x} + 1} = x - \tan{\frac{x}{2}}+C


Integrands involving only cosecant[edit]

\int csc (ax) \mathrm{d}x= -\frac{1}{a}\ln{\left| \csc{ax}+\cot{ax}\right|}+C
\int \csc^2{x} \, \mathrm{d}x = -\cot{x}+C
\int \csc^n{ax} \, \mathrm{d}x = -\frac{\csc^{n-1}\left(ax\right)\cos\left(ax\right)}{a(n-1)} \,+\, \frac{n-2}{n-1}\int \csc^{n-2}{ax} \, \mathrm{d}x \qquad \mbox{ (for }n \ne 1\mbox{)}\,\!
\int \frac{\mathrm{d}x}{\csc{x} + 1} = x - \frac{2\sin{\frac{x}{2}}}{\cos{\frac{x}{2}}+\sin{\frac{x}{2}}}+C
\int \frac{\mathrm{d}x}{\csc{x} - 1} = \frac{2\sin{\frac{x}{2}}}{\cos{\frac{x}{2}}-\sin{\frac{x}{2}}}-x+C

Integrands involving only cotangent[edit]

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

Integrands involving both sine and cosine[edit]

\int\frac{\mathrm{d}x}{\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{\mathrm{d}x}{(\cos ax\pm\sin ax)^2} = \frac{1}{2a}\tan\left(ax\mp\frac{\pi}{4}\right)+C
\int\frac{\mathrm{d}x}{(\cos x + \sin x)^n} = \frac{1}{n-1}\left(\frac{\sin x - \cos x}{(\cos x + \sin x)^{n - 1}} - 2(n - 2)\int\frac{\mathrm{d}x}{(\cos x + \sin x)^{n-2}} \right)
\int\frac{\cos ax\;\mathrm{d}x}{\cos ax + \sin ax} = \frac{x}{2} + \frac{1}{2a}\ln\left|\sin ax + \cos ax\right|+C
\int\frac{\cos ax\;\mathrm{d}x}{\cos ax - \sin ax} = \frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax - \cos ax\right|+C
\int\frac{\sin ax\;\mathrm{d}x}{\cos ax + \sin ax} = \frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax + \cos ax\right|+C
\int\frac{\sin ax\;\mathrm{d}x}{\cos ax - \sin ax} = -\frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax - \cos ax\right|+C
\int\frac{\cos ax\;\mathrm{d}x}{\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\;\mathrm{d}x}{\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\;\mathrm{d}x}{\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\;\mathrm{d}x}{\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\;\mathrm{d}x = -\frac{1}{2a}\cos^2 ax +C\,\!
\int\sin a_1x\cos a_2x\;\mathrm{d}x = -\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\;\mathrm{d}x = \frac{1}{a(n+1)}\sin^{n+1} ax +C\qquad\mbox{(for }n\neq -1\mbox{)}\,\!
\int\sin ax\cos^n ax\;\mathrm{d}x = -\frac{1}{a(n+1)}\cos^{n+1} ax +C\qquad\mbox{(for }n\neq -1\mbox{)}\,\!
\int\sin^n ax\cos^m ax\;\mathrm{d}x = -\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\;\mathrm{d}x  \qquad\mbox{(for }m,n>0\mbox{)}\,\!
also: \int\sin^n ax\cos^m ax\;\mathrm{d}x = \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\;\mathrm{d}x \qquad\mbox{(for }m,n>0\mbox{)}\,\!
\int\frac{\mathrm{d}x}{\sin ax\cos ax} = \frac{1}{a}\ln\left|\tan ax\right|+C
\int\frac{\mathrm{d}x}{\sin ax\cos^n ax} = \frac{1}{a(n-1)\cos^{n-1} ax}+\int\frac{\mathrm{d}x}{\sin ax\cos^{n-2} ax} \qquad\mbox{(for }n\neq 1\mbox{)}\,\!
\int\frac{\mathrm{d}x}{\sin^n ax\cos ax} = -\frac{1}{a(n-1)\sin^{n-1} ax}+\int\frac{\mathrm{d}x}{\sin^{n-2} ax\cos ax} \qquad\mbox{(for }n\neq 1\mbox{)}\,\!
\int\frac{\sin ax\;\mathrm{d}x}{\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\;\mathrm{d}x}{\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\;\mathrm{d}x}{\cos^n ax} = \frac{\sin ax}{a(n-1)\cos^{n-1}ax}-\frac{1}{n-1}\int\frac{\mathrm{d}x}{\cos^{n-2}ax} \qquad\mbox{(for }n\neq 1\mbox{)}\,\!
\int\frac{\sin^n ax\;\mathrm{d}x}{\cos ax} = -\frac{\sin^{n-1} ax}{a(n-1)} + \int\frac{\sin^{n-2} ax\;\mathrm{d}x}{\cos ax} \qquad\mbox{(for }n\neq 1\mbox{)}\,\!
\int\frac{\sin^n ax\;\mathrm{d}x}{\cos^m ax} = \frac{\sin^{n+1} ax}{a(m-1)\cos^{m-1} ax}-\frac{n-m+2}{m-1}\int\frac{\sin^n ax\;\mathrm{d}x}{\cos^{m-2} ax} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!
also: \int\frac{\sin^n ax\;\mathrm{d}x}{\cos^m ax} = -\frac{\sin^{n-1} ax}{a(n-m)\cos^{m-1} ax}+\frac{n-1}{n-m}\int\frac{\sin^{n-2} ax\;\mathrm{d}x}{\cos^m ax} \qquad\mbox{(for }m\neq n\mbox{)}\,\!
also: \int\frac{\sin^n ax\;\mathrm{d}x}{\cos^m ax} = \frac{\sin^{n-1} ax}{a(m-1)\cos^{m-1} ax}-\frac{n-1}{m-1}\int\frac{\sin^{n-2} ax\;\mathrm{d}x}{\cos^{m-2} ax} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!
\int\frac{\cos ax\;\mathrm{d}x}{\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\;\mathrm{d}x}{\sin ax} = \frac{1}{a}\left(\cos ax+\ln\left|\tan\frac{ax}{2}\right|\right) +C
\int\frac{\cos^2 ax\;\mathrm{d}x}{\sin^n ax} = -\frac{1}{n-1}\left(\frac{\cos ax}{a\sin^{n-1} ax)}+\int\frac{\mathrm{d}x}{\sin^{n-2} ax}\right) \qquad\mbox{(for }n\neq 1\mbox{)}
\int\frac{\cos^n ax\;\mathrm{d}x}{\sin^m ax} = -\frac{\cos^{n+1} ax}{a(m-1)\sin^{m-1} ax} - \frac{n-m+2}{m-1}\int\frac{\cos^n ax\;\mathrm{d}x}{\sin^{m-2} ax} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!
also: \int\frac{\cos^n ax\;\mathrm{d}x}{\sin^m ax} = \frac{\cos^{n-1} ax}{a(n-m)\sin^{m-1} ax} + \frac{n-1}{n-m}\int\frac{\cos^{n-2} ax\;\mathrm{d}x}{\sin^m ax} \qquad\mbox{(for }m\neq n\mbox{)}\,\!
also: \int\frac{\cos^n ax\;\mathrm{d}x}{\sin^m ax} = -\frac{\cos^{n-1} ax}{a(m-1)\sin^{m-1} ax} - \frac{n-1}{m-1}\int\frac{\cos^{n-2} ax\;\mathrm{d}x}{\sin^{m-2} ax} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!

Integrands involving both sine and tangent[edit]

\int \sin ax \tan ax\;\mathrm{d}x = \frac{1}{a}(\ln|\sec ax + \tan ax| - \sin ax)+C\,\!
\int\frac{\tan^n ax\;\mathrm{d}x}{\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[edit]

\int\frac{\tan^n ax\;\mathrm{d}x}{\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[edit]

\int\frac{\cot^n ax\;\mathrm{d}x}{\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[edit]

\int\frac{\cot^n ax\;\mathrm{d}x}{\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[edit]

 \int\sec x \tan x\;\mathrm{d}x= \sec x + C

Integrals with symmetric limits[edit]

\int_{{-c}}^{{c}}\sin {x}\;\mathrm{d}x = 0 \!
\int_{{-c}}^{{c}}\cos {x}\;\mathrm{d}x = 2\int_{{0}}^{{c}}\cos {x}\;\mathrm{d}x = 2\int_{{-c}}^{{0}}\cos {x}\;\mathrm{d}x = 2\sin {c} \!
\int_{{-c}}^{{c}}\tan {x}\;\mathrm{d}x = 0 \!
\int_{-\frac{a}{2}}^{\frac{a}{2}} x^2\cos^2 {\frac{n\pi x}{a}}\;\mathrm{d}x = \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}}\;\mathrm{d}x = \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[edit]

\int_{{0}}^{{2 \pi}}\sin^{2m+1}{x}\cos^{2n+1}{x}\;\mathrm{d}x = 0 \! \qquad \{n,m\} \in \mathbb{Z}

See also[edit]

References[edit]