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

\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|\csc {ax}+\cot {ax}\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

\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_{1}x\cos a_{2}x\;{\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

\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

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

\int \frac{\mathrm{d}x}{\sec{x} - 1} = - x - \cot{\frac{x}{2}}+C

Integrands involving only cosecant

\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^3{x} \, dx = -\frac{1}{2}\csc x \cot x - \frac{1}{2}\ln|\csc x + \cot x| + C.

\int \csc^n{ax} \, \mathrm{d}x = -\frac{\csc^{n-2}{ax} \cot{ax}}{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}{\cot{\frac{x}{2}}+1}+C

\int \frac{\mathrm{d}x}{\csc{x} - 1} = - x + \frac{2}{\cot{\frac{x}{2}}-1}+C

Integrands involving only cotangent

\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

\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

\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

\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

\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

\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

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

Integrand involving with cosecant and cotangent

\int \csc x\cot x\;\mathrm {d} x=-\csc x+C

Integrals in a quarter period

\int _{0}^{\frac {\pi }{2}}\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}x\,dx={\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}\;\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

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

References

Lists of integrals

Rational functions Irrational functions Trigonometric functions Inverse trigonometric functions Hyperbolic functions Inverse hyperbolic functions Exponential functions Logarithmic functions Gaussian functions

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List of integrals of trigonometric functions

Integrands involving only sine

Integrands involving only cosine

Integrands involving only tangent

Integrands involving only secant

Integrands involving only cosecant

Integrands involving only cotangent

Integrands involving both sine and cosine

Integrands involving both sine and tangent

Integrand involving both cosine and tangent

Integrand involving both sine and cotangent

Integrand involving both cosine and cotangent

Integrand involving both secant and tangent

Integrand involving with cosecant and cotangent

Integrals in a quarter period

Integrals with symmetric limits

Integral over a full circle

See also

References