List of integrals of trigonometric functions

Template:Short description Template:More sources needed Template:Trigonometry 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,

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

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

Integrands involving only sine

• $\int \sin axdx=-\frac{1}{a}\cos ax+C$
• $\int {\sin }^{2}axdx=\frac{x}{2}-\frac{1}{4a}\sin 2ax+C=\frac{x}{2}-\frac{1}{2a}\sin ax\cos ax+C$
• $\int {\sin }^{3}axdx=\frac{\cos 3ax}{12a}-\frac{3\cos ax}{4a}+C$
• $\int x{\sin }^{2}axdx=\frac{x^2}{4}-\frac{x}{4a}\sin 2ax-\frac{1}{8a^2}\cos 2ax+C$
• $\int x^2{\sin }^{2}axdx=\frac{x^3}{6}-(\frac{x^2}{4a}-\frac{1}{8a^3})\sin 2ax-\frac{x}{4a^2}\cos 2ax+C$
• $\int x\sin axdx=\frac{\sin ax}{a^2}-\frac{x\cos ax}{a}+C$
• $\int (\sin b_{1}x)(\sin b_{2}x)dx=\frac{\sin ((b_2-b_1)x)}{2(b_2-b_1)}-\frac{\sin ((b_1+b_2)x)}{2(b_1+b_2)}+C\text{(for }|b_1|\ne |b_2|\text{)}$
• $\int {\sin }^{n}axdx=-\frac{{\sin }^{n-1}ax\cos ax}{na}+\frac{n-1}{n}\int {\sin }^{n-2}axdx\text{(for }n>0\text{)}$
• $\int \frac{dx}{\sin ax}=-\frac{1}{a}\ln |\csc ax+\cot ax|+C$
• $\int \frac{dx}{{\sin }^{n}ax}=\frac{\cos ax}{a(1-n){\sin }^{n-1}ax}+\frac{n-2}{n-1}\int \frac{dx}{{\sin }^{n-2}ax}\text{(for }n>1\text{)}$
• $\begin{array}{rl}\int x^n\sin axdx& =-\frac{x^n}{a}\cos ax+\frac{n}{a}\int x^{n-1}\cos axdx\\ & =\sum _{k=0}^{2k\le n}(-1{)}^{k+1}\frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!}\cos ax+\sum _{k=0}^{2k+1\le n}(-1{)}^k\frac{x^{n-1-2k}}{a^{2+2k}}\frac{n!}{(n-2k-1)!}\sin ax\\ & =-\sum _{k=0}^n\frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\cos (ax+k\frac{\pi }{2})\text{(for }n>0\text{)}\end{array}$
• $\int \frac{\sin ax}{x}dx=\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n\frac{(ax{)}^{2n+1}}{(2n+1)\cdot (2n+1)!}+C$
• $\int \frac{\sin ax}{x^n}dx=-\frac{\sin ax}{(n-1)x^{n-1}}+\frac{a}{n-1}\int \frac{\cos ax}{x^{n-1}}dx$
• $\int \sin \mathrm{(}{ax}^2\mathrm{+}bx\mathrm{+}c\mathrm{)}dx\mathrm{=}\{\begin{array}{rl}& \sqrt{a}\sqrt{\frac{{\pi }}{2}}\cos \left(\frac{b^2\mathrm{-}4ac}{4a}\right)S\left(\frac{2ax\mathrm{+}b}{\sqrt{2a{\pi }}}\right)\mathrm{+}\sqrt{a}\sqrt{\frac{{\pi }}{2}}\sin \left(\frac{b^2\mathrm{-}4ac}{4a}\right)C\left(\frac{2ax\mathrm{+}b}{\sqrt{2a{\pi }}}\right)to{b}^2\mathrm{-}4ac>0\\ & \sqrt{a}\sqrt{\frac{{\pi }}{2}}\cos \left(\frac{b^2\mathrm{-}4ac}{4a}\right)S\left(\frac{2ax\mathrm{+}b}{\sqrt{2a{\pi }}}\right)\mathrm{-}\sqrt{a}\sqrt{\frac{{\pi }}{2}}\sin \left(\frac{b^2\mathrm{-}4ac}{4a}\right)C\left(\frac{2ax\mathrm{+}b}{\sqrt{2a{\pi }}}\right)to{b}^2\mathrm{-}4ac<0\end{array}fora╱\mathrm{=}0,a>0$
• $\int \frac{dx}{1\pm \sin ax}=\frac{1}{a}\tan (\frac{ax}{2}\mp \frac{\pi }{4})+C$
• $\int \frac{xdx}{1+\sin ax}=\frac{x}{a}\tan (\frac{ax}{2}-\frac{\pi }{4})+\frac{2}{a^2}\ln |\cos (\frac{ax}{2}-\frac{\pi }{4})|+C$
• $\int \frac{xdx}{1-\sin ax}=\frac{x}{a}\cot (\frac{\pi }{4}-\frac{ax}{2})+\frac{2}{a^2}\ln |\sin (\frac{\pi }{4}-\frac{ax}{2})|+C$
• $\int \frac{\sin axdx}{1\pm \sin ax}=\pm x+\frac{1}{a}\tan (\frac{\pi }{4}\mp \frac{ax}{2})+C$

Integrands involving only cosine

• $\int \cos axdx=\frac{1}{a}\sin ax+C$
• $\int {\cos }^{2}axdx=\frac{x}{2}+\frac{1}{4a}\sin 2ax+C=\frac{x}{2}+\frac{1}{2a}\sin ax\cos ax+C$
• $\int {\cos }^{n}axdx=\frac{{\cos }^{n-1}ax\sin ax}{na}+\frac{n-1}{n}\int {\cos }^{n-2}axdx\text{(for }n>0\text{)}$
• $\int x\cos axdx=\frac{\cos ax}{a^2}+\frac{x\sin ax}{a}+C$
• $\int x^2{\cos }^{2}axdx=\frac{x^3}{6}+(\frac{x^2}{4a}-\frac{1}{8a^3})\sin 2ax+\frac{x}{4a^2}\cos 2ax+C$
• $\begin{array}{rl}\int x^n\cos axdx& =\frac{x^n\sin ax}{a}-\frac{n}{a}\int x^{n-1}\sin axdx\\ & =\sum _{k=0}^{2k+1\le n}(-1{)}^k\frac{x^{n-2k-1}}{a^{2+2k}}\frac{n!}{(n-2k-1)!}\cos ax+\sum _{k=0}^{2k\le n}(-1{)}^k\frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!}\sin ax\\ & =\sum _{k=0}^n(-1{)}^{\lfloor k/2\rfloor }\frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\cos (ax-\frac{(-1{)}^k+1}{2}\frac{\pi }{2})\\ & =\sum _{k=0}^n\frac{x^{n-k}}{a^{1+k}}\frac{n!}{(n-k)!}\sin (ax+k\frac{\pi }{2})\text{(for }n>0\text{)}\end{array}$
• $\int \frac{\cos ax}{x}dx=\ln |ax|+\sum _{k=1}^{\mathrm{\infty }}(-1{)}^k\frac{(ax{)}^{2k}}{2k\cdot (2k)!}+C$
• $\int \frac{\cos ax}{x^n}dx=-\frac{\cos ax}{(n-1)x^{n-1}}-\frac{a}{n-1}\int \frac{\sin ax}{x^{n-1}}dx\text{(for }n\ne 1\text{)}$
• $\int \frac{dx}{\cos ax}=\frac{1}{a}\ln |\tan (\frac{ax}{2}+\frac{\pi }{4})|+C$
• $\int \frac{dx}{{\cos }^{n}ax}=\frac{\sin ax}{a(n-1){\cos }^{n-1}ax}+\frac{n-2}{n-1}\int \frac{dx}{{\cos }^{n-2}ax}\text{(for }n>1\text{)}$
• $\int \frac{dx}{1+\cos ax}=\frac{1}{a}\tan \frac{ax}{2}+C$
• $\int \frac{dx}{1-\cos ax}=-\frac{1}{a}\cot \frac{ax}{2}+C$
• $\int \frac{xdx}{1+\cos ax}=\frac{x}{a}\tan \frac{ax}{2}+\frac{2}{a^2}\ln |\cos \frac{ax}{2}|+C$
• $\int \frac{xdx}{1-\cos ax}=-\frac{x}{a}\cot \frac{ax}{2}+\frac{2}{a^2}\ln |\sin \frac{ax}{2}|+C$
• $\int \frac{\cos axdx}{1+\cos ax}=x-\frac{1}{a}\tan \frac{ax}{2}+C$
• $\int \frac{\cos axdx}{1-\cos ax}=-x-\frac{1}{a}\cot \frac{ax}{2}+C$
• $\int (\cos a_{1}x)(\cos a_{2}x)dx=\frac{\sin ((a_2-a_1)x)}{2(a_2-a_1)}+\frac{\sin ((a_2+a_1)x)}{2(a_2+a_1)}+C\text{(for }|a_1|\ne |a_2|\text{)}$

Integrands involving only tangent

• $\int \tan axdx=-\frac{1}{a}\ln |\cos ax|+C=\frac{1}{a}\ln |\sec ax|+C$
• $\int {\tan }^{2}xdx=\tan x-x+C$
• $\int {\tan }^{n}axdx=\frac{1}{a(n-1)}{\tan }^{n-1}ax-\int {\tan }^{n-2}axdx\text{(for }n\ne 1\text{)}$
• $\int \frac{dx}{q\tan ax+p}=\frac{1}{p^2+q^2}(px+\frac{q}{a}\ln |q\sin ax+p\cos ax|)+C\text{(for }{p}^2+q^2\ne 0\text{)}$
• $\int \frac{dx}{\tan ax\pm 1}=\pm \frac{x}{2}+\frac{1}{2a}\ln |\sin ax\pm \cos ax|+C$
• $\int \frac{\tan axdx}{\tan ax\pm 1}=\frac{x}{2}\mp \frac{1}{2a}\ln |\sin ax\pm \cos ax|+C$

Integrands involving only secant

• $\int \sec axdx=\frac{1}{a}\ln |\sec ax+\tan ax|+C=\frac{1}{a}\ln |\tan (\frac{ax}{2}+\frac{\pi }{4})|+C=\frac{1}{a}\mathrm{artanh}(\sin ax)+C$
• $\int {\sec }^{2}xdx=\tan x+C$
• $\int {\sec }^{3}xdx=\frac{1}{2}\sec x\tan x+\frac{1}{2}\ln |\sec x+\tan x|+C.$
• $\int {\sec }^{n}axdx=\frac{{\sec }^{n-2}ax\tan ax}{a(n-1)}+\frac{n-2}{n-1}\int {\sec }^{n-2}axdx\text{ (for }n\ne 1\text{)}$
• $\int \frac{dx}{\sec x+1}=x-\tan \frac{x}{2}+C$
• $\int \frac{dx}{\sec x-1}=-x-\cot \frac{x}{2}+C$

Integrands involving only cosecant

• $\int \csc axdx=-\frac{1}{a}\ln |\csc ax+\cot ax|+C=\frac{1}{a}\ln |\csc ax-\cot ax|+C=\frac{1}{a}\ln |\tan \left(\frac{ax}{2}\right)|+C$
• $\int {\csc }^{2}xdx=-\cot x+C$
• $\int {\csc }^{3}xdx=-\frac{1}{2}\csc x\cot x-\frac{1}{2}\ln |\csc x+\cot x|+C=-\frac{1}{2}\csc x\cot x+\frac{1}{2}\ln |\csc x-\cot x|+C$
• $\int {\csc }^{n}axdx=-\frac{{\csc }^{n-2}ax\cot ax}{a(n-1)}+\frac{n-2}{n-1}\int {\csc }^{n-2}axdx\text{ (for }n\ne 1\text{)}$
• $\int \frac{dx}{\csc x+1}=x-\frac{2}{\cot \frac{x}{2}+1}+C$
• $\int \frac{dx}{\csc x-1}=-x+\frac{2}{\cot \frac{x}{2}-1}+C$

Integrands involving only cotangent

• $\int \cot axdx=\frac{1}{a}\ln |\sin ax|+C$
• $\int {\cot }^{2}xdx=-\cot x-x+C$
• $\int {\cot }^{n}axdx=-\frac{1}{a(n-1)}{\cot }^{n-1}ax-\int {\cot }^{n-2}axdx\text{(for }n\ne 1\text{)}$
• $\int \frac{dx}{1+\cot ax}=\int \frac{\tan axdx}{\tan ax+1}=\frac{x}{2}-\frac{1}{2a}\ln |\sin ax+\cos ax|+C$
• $\int \frac{dx}{1-\cot ax}=\int \frac{\tan axdx}{\tan ax-1}=\frac{x}{2}+\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

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.

• $\int \frac{dx}{\cos ax\pm \sin ax}=\frac{1}{a\sqrt{2}}\ln |\tan (\frac{ax}{2}\pm \frac{\pi }{8})|+C$
• $\int \frac{dx}{(\cos ax\pm \sin ax{)}^2}=\frac{1}{2a}\tan (ax\mp \frac{\pi }{4})+C$
• $\int \frac{dx}{(\cos x+\sin x{)}^n}=\frac{1}{2(n-1)}(\frac{\sin x-\cos x}{(\cos x+\sin x{)}^{n-1}}+(n-2)\int \frac{dx}{(\cos x+\sin x{)}^{n-2}})$
• $\int \frac{\cos axdx}{\cos ax+\sin ax}=\frac{x}{2}+\frac{1}{2a}\ln |\sin ax+\cos ax|+C$
• $\int \frac{\cos axdx}{\cos ax-\sin ax}=\frac{x}{2}-\frac{1}{2a}\ln |\sin ax-\cos ax|+C$
• $\int \frac{\sin axdx}{\cos ax+\sin ax}=\frac{x}{2}-\frac{1}{2a}\ln |\sin ax+\cos ax|+C$
• $\int \frac{\sin axdx}{\cos ax-\sin ax}=-\frac{x}{2}-\frac{1}{2a}\ln |\sin ax-\cos ax|+C$
• $\int \frac{\cos axdx}{(\sin ax)(1+\cos ax)}=-\frac{1}{4a}{\tan }^2\frac{ax}{2}+\frac{1}{2a}\ln |\tan \frac{ax}{2}|+C$
• $\int \frac{\cos axdx}{(\sin ax)(1-\cos ax)}=-\frac{1}{4a}{\cot }^2\frac{ax}{2}-\frac{1}{2a}\ln |\tan \frac{ax}{2}|+C$
• $\int \frac{\sin axdx}{(\cos ax)(1+\sin ax)}=\frac{1}{4a}{\cot }^2(\frac{ax}{2}+\frac{\pi }{4})+\frac{1}{2a}\ln |\tan (\frac{ax}{2}+\frac{\pi }{4})|+C$
• $\int \frac{\sin axdx}{(\cos ax)(1-\sin ax)}=\frac{1}{4a}{\tan }^2(\frac{ax}{2}+\frac{\pi }{4})-\frac{1}{2a}\ln |\tan (\frac{ax}{2}+\frac{\pi }{4})|+C$
• $\int (\sin ax)(\cos ax)dx=\frac{1}{2a}{\sin }^{}ax+C$
• $\int (\sin a_{1}x)(\cos a_{2}x)dx=-\frac{\cos ((a_1-a_2)x)}{2(a_1-a_2)}-\frac{\cos ((a_1+a_2)x)}{2(a_1+a_2)}+C\text{(for }|a_1|\ne |a_2|\text{)}$
• $\int ({\sin }^{n}ax)(\cos ax)dx=\frac{1}{a(n+1)}{\sin }^{}ax+C\text{(for }n\ne -1\text{)}$
• $\int (\sin ax)({\cos }^{}ax)dx=-\frac{1}{a(n+1)}{\cos }^{}ax+C\text{(for }n\ne -1\text{)}$
• $\begin{array}{rl}\int ({\sin }^{n}ax)({\cos }^{m}ax)dx& =-\frac{({\sin }^{n-1}ax)({\cos }^{m+1}ax)}{a(n+m)}+\frac{n-1}{n+m}\int ({\sin }^{n-2}ax)({\cos }^{m}ax)dx\text{(for }m,n>0\text{)}\\ & =\frac{({\sin }^{n+1}ax)({\cos }^{m-1}ax)}{a(n+m)}+\frac{m-1}{n+m}\int ({\sin }^{n}ax)({\cos }^{m-2}ax)dx\text{(for }m,n>0\text{)}\end{array}$
• $\int \frac{dx}{(\sin ax)(\cos ax)}=\frac{1}{a}\ln |\tan ax|+C$
• $\int \frac{dx}{(\sin ax)({\cos }^{}ax)}=\frac{1}{a(n-1){\cos }^{n-1}ax}+\int \frac{dx}{(\sin ax)({\cos }^{}ax)}\text{(for }n\ne 1\text{)}$
• $\int \frac{dx}{({\sin }^{n}ax)(\cos ax)}=-\frac{1}{a(n-1){\sin }^{n-1}ax}+\int \frac{dx}{({\sin }^{n-2}ax)(\cos ax)}\text{(for }n\ne 1\text{)}$
• $\int \frac{\sin axdx}{{\cos }^{n}ax}=\frac{1}{a(n-1){\cos }^{n-1}ax}+C\text{(for }n\ne 1\text{)}$
• $\int \frac{{\sin }^{2}axdx}{\cos ax}=-\frac{1}{a}\sin ax+\frac{1}{a}\ln |\tan (\frac{\pi }{4}+\frac{ax}{2})|+C$
• $\int \frac{{\sin }^{2}axdx}{{\cos }^{n}ax}=\frac{\sin ax}{a(n-1){\cos }^{n-1}ax}-\frac{1}{n-1}\int \frac{dx}{{\cos }^{n-2}ax}\text{(for }n\ne 1\text{)}$
• $\begin{array}{rl}\int \frac{{\sin }^{2}x}{1+{\cos }^{2}x}dx& =\sqrt{2}\mathrm{arctangant}\left(\frac{\tan x}{\sqrt{2}}\right)-x\text{(for x in}]-\frac{\pi }{2};+\frac{\pi }{2}[\text{)}\\ & =\sqrt{2}\mathrm{arctangant}\left(\frac{\tan x}{\sqrt{2}}\right)-\mathrm{arctangant}(\tan x)\text{(this time x being any real number }\text{)}\end{array}$
• $\int \frac{{\sin }^{n}axdx}{\cos ax}=-\frac{{\sin }^{n-1}ax}{a(n-1)}+\int \frac{{\sin }^{n-2}axdx}{\cos ax}\text{(for }n\ne 1\text{)}$
• $\int \frac{{\sin }^{n}axdx}{{\cos }^{m}ax}=\{\begin{array}{ll}\frac{{\sin }^{n+1}ax}{a(m-1){\cos }^{m-1}ax}-\frac{n-m+2}{m-1}\int \frac{{\sin }^{n}axdx}{{\cos }^{m-2}ax}& \text{(for }m\ne 1\text{)}\\ \frac{{\sin }^{n-1}ax}{a(m-1){\cos }^{m-1}ax}-\frac{n-1}{m-1}\int \frac{{\sin }^{n-2}axdx}{{\cos }^{m-2}ax}& \text{(for }m\ne 1\text{)}\\ -\frac{{\sin }^{n-1}ax}{a(n-m){\cos }^{m-1}ax}+\frac{n-1}{n-m}\int \frac{{\sin }^{n-2}axdx}{{\cos }^{m}ax}& \text{(for }m\ne n\text{)}\end{array}$
• $\int \frac{\cos axdx}{{\sin }^{n}ax}=-\frac{1}{a(n-1){\sin }^{n-1}ax}+C\text{(for }n\ne 1\text{)}$
• $\int \frac{{\cos }^{2}axdx}{\sin ax}=\frac{1}{a}(\cos ax+\ln |\tan \frac{ax}{2}|)+C$
• $\int \frac{{\cos }^{2}axdx}{{\sin }^{n}ax}=-\frac{1}{n-1}(\frac{\cos ax}{a{\sin }^{n-1}ax}+\int \frac{dx}{{\sin }^{n-2}ax})\text{(for }n\ne 1\text{)}$
• $\int \frac{{\cos }^{n}axdx}{{\sin }^{m}ax}=\{\begin{array}{ll}-\frac{{\cos }^{n+1}ax}{a(m-1){\sin }^{m-1}ax}-\frac{n-m+2}{m-1}\int \frac{{\cos }^{n}axdx}{{\sin }^{m-2}ax}& \text{(for }n\ne 1\text{)}\\ -\frac{{\cos }^{n-1}ax}{a(m-1){\sin }^{m-1}ax}-\frac{n-1}{m-1}\int \frac{{\cos }^{n-2}axdx}{{\sin }^{m-2}ax}& \text{(for }m\ne 1\text{)}\\ \frac{{\cos }^{n-1}ax}{a(n-m){\sin }^{m-1}ax}+\frac{n-1}{n-m}\int \frac{{\cos }^{n-2}axdx}{{\sin }^{m}ax}& \text{(for }m\ne n\text{)}\end{array}$

Integrands involving both sine and tangent

• $\int (\sin ax)(\tan ax)dx=\frac{1}{a}(\ln |\sec ax+\tan ax|-\sin ax)+C$
• $\int \frac{{\tan }^{n}axdx}{{\sin }^{2}ax}=\frac{1}{a(n-1)}{\tan }^{n-1}(ax)+C\text{(for }n\ne 1\text{)}$

Integrand involving both cosine and tangent

• $\int \frac{{\tan }^{n}axdx}{{\cos }^{2}ax}=\frac{1}{a(n+1)}{\tan }^{n+1}ax+C\text{(for }n\ne -1\text{)}$

Integrand involving both sine and cotangent

• $\int \frac{{\cot }^{n}axdx}{{\sin }^{2}ax}=-\frac{1}{a(n+1)}{\cot }^{n+1}ax+C\text{(for }n\ne -1\text{)}$

Integrand involving both cosine and cotangent

• $\int \frac{{\cot }^{n}axdx}{{\cos }^{2}ax}=\frac{1}{a(1-n)}{\tan }^{1-n}ax+C\text{(for }n\ne 1\text{)}$

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 Template:Startplainlist

• $\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\sin }^{n}xdx=\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\cos }^{n}xdx=\frac{1}{2}B(\frac{n+1}{2},\frac{1}{2})=\{\begin{array}{ll}\frac{n-1}{n}\cdot \frac{n-3}{n-2}\cdots \frac{3}{4}\cdot \frac{1}{2}\cdot \frac{\pi }{2},& \text{if }n\text{ is even}\\ \frac{n-1}{n}\cdot \frac{n-3}{n-2}\cdots \frac{4}{5}\cdot \frac{2}{3},& \text{if }n\text{ is odd and more than 1}\\ 1,& \text{if }n=1\end{array}$

Using the modified Struve functions $L_{\alpha }(x)$ and modified Bessel functions $I_{\alpha }(x)$ one can write Template:Startplainlist

• $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\exp (k\cdot \sin (x))dx=\frac{\pi }{2}(I_0(k)+L_0(k))$

Integrals with symmetric limits

• $\underset{-c}{\overset{c}{\int }}\sin xdx=0$
• $\underset{-c}{\overset{c}{\int }}\cos xdx=2\underset{0}{\overset{c}{\int }}\cos xdx=2\underset{-c}{\overset{0}{\int }}\cos xdx=2\sin c$
• $\underset{-c}{\overset{c}{\int }}\tan xdx=0$
• $\underset{-\frac{a}{2}}{\overset{\frac{a}{2}}{\int }}{x}^2{\cos }^2\frac{n\pi x}{a}dx=\frac{a^3(n^2{\pi }^2-6)}{24n^2{\pi }^2}\text{(for }n=1,3,5...\text{)}$
• $\underset{\frac{-a}{2}}{\overset{\frac{a}{2}}{\int }}{x}^2{\sin }^2\frac{n\pi x}{a}dx=\frac{a^3(n^2{\pi }^2-6(-1{)}^n)}{24n^2{\pi }^2}=\frac{a^3}{24}(1-6\frac{(-1{)}^n}{n^2{\pi }^2})\text{(for }n=1,2,3,...\text{)}$

Integral over a full circle

• $\underset{0}{\overset{2\pi }{\int }}{\sin }^{2m+1}x{\cos }^{n}xdx=0n,m\in \mathbb{\mathbb{Z}}$
• $\underset{0}{\overset{2\pi }{\int }}{\sin }^{m}x{\cos }^{2n+1}xdx=0n,m\in \mathbb{\mathbb{Z}}$

See also

• Trigonometric integral