List of integrals of rational functions

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. See also 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.

Integrals involving only sine

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

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

$\int {\sin }^{3}ax\mathrm{d}x=\frac{\cos 3ax}{12a}-\frac{3\cos ax}{4a}+C$

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

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

$\int \sin b_{1}x\sin b_{2}x\mathrm{d}x=\frac{\sin ((b_2-b_1)x)}{2(b_2-b_1)}-\frac{\sin ((b_1+b_2)x)}{2(b_1+b_2)}+C\text{(for }\left|b_1\right|\ne \left|b_2\right|\text{)}$

$\int {\sin }^{n}ax\mathrm{d}x=-\frac{{\sin }^{n-1}ax\cos ax}{na}+\frac{n-1}{n}\int {\sin }^{n-2}ax\mathrm{d}x\text{(for }n>2\text{)}$

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

$\int \frac{\mathrm{d}x}{{\sin }^{n}ax}=\frac{\cos ax}{a(1-n){\sin }^{n-1}ax}+\frac{n-2}{n-1}\int \frac{\mathrm{d}x}{{\sin }^{n-2}ax}\text{(for }n>1\text{)}$

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

$\int x^n\sin ax\mathrm{d}x=-\frac{x^n}{a}\cos ax+\frac{n}{a}\int x^{n-1}\cos ax\mathrm{d}x=\sum _{k=0}^{2k\le n}(-1{)}^{k+1}\frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!}\cos ax+\sum _{k=0}^{2k+1\le n}(-1{)}^k\frac{x^{n-1-2k}}{a^{2+2k}}\frac{n!}{(n-2k-1)!}\sin ax\text{(for }n>0\text{)}$

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

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

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

$\int \frac{x\mathrm{d}x}{1+\sin ax}=\frac{x}{a}\tan (\frac{ax}{2}-\frac{\pi }{4})+\frac{2}{a^2}\ln |\cos (\frac{ax}{2}-\frac{\pi }{4})|+C$

$\int \frac{x\mathrm{d}x}{1-\sin ax}=\frac{x}{a}\cot (\frac{\pi }{4}-\frac{ax}{2})+\frac{2}{a^2}\ln |\sin (\frac{\pi }{4}-\frac{ax}{2})|+C$

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

Integrands involving only cosine

$\int \cos ax\mathrm{d}x=\frac{1}{a}\sin ax+C$

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

$\int {\cos }^{n}ax\mathrm{d}x=\frac{{\cos }^{n-1}ax\sin ax}{na}+\frac{n-1}{n}\int {\cos }^{n-2}ax\mathrm{d}x\text{(for }n>0\text{)}$

$\int x\cos ax\mathrm{d}x=\frac{\cos ax}{a^2}+\frac{x\sin ax}{a}+C$

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

$\int x^n\cos ax\mathrm{d}x=\frac{x^n\sin ax}{a}-\frac{n}{a}\int x^{n-1}\sin ax\mathrm{d}x=\sum _{k=0}^{2k+1\le n}(-1{)}^k\frac{x^{n-2k-1}}{a^{2+2k}}\frac{n!}{(n-2k-1)!}\cos ax+\sum _{k=0}^{2k\le n}(-1{)}^k\frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!}\sin ax$

$\int \frac{\cos ax}{x}\mathrm{d}x=\ln \left|ax\right|+\sum _{k=1}^{\mathrm{\infty }}(-1{)}^k\frac{(ax{)}^{2k}}{2k\cdot (2k)!}+C$

$\int \frac{\cos ax}{x^n}\mathrm{d}x=-\frac{\cos ax}{(n-1)x^{n-1}}-\frac{a}{n-1}\int \frac{\sin ax}{x^{n-1}}\mathrm{d}x\text{(for }n\ne 1\text{)}$

$\int \frac{\mathrm{d}x}{\cos ax}=\frac{1}{a}\ln |\tan (\frac{ax}{2}+\frac{\pi }{4})|+C$

$\int \frac{\mathrm{d}x}{{\cos }^{n}ax}=\frac{\sin ax}{a(n-1){\cos }^{n-1}ax}+\frac{n-2}{n-1}\int \frac{\mathrm{d}x}{{\cos }^{n-2}ax}\text{(for }n>1\text{)}$

$\int \frac{\mathrm{d}x}{1+\cos ax}=\frac{1}{a}\tan \frac{ax}{2}+C$

$\int \frac{\mathrm{d}x}{1-\cos ax}=-\frac{1}{a}\cot \frac{ax}{2}+C$

$\int \frac{x\mathrm{d}x}{1+\cos ax}=\frac{x}{a}\tan \frac{ax}{2}+\frac{2}{a^2}\ln |\cos \frac{ax}{2}|+C$

$\int \frac{x\mathrm{d}x}{1-\cos ax}=-\frac{x}{a}\cot \frac{ax}{2}+\frac{2}{a^2}\ln |\sin \frac{ax}{2}|+C$

$\int \frac{\cos ax\mathrm{d}x}{1+\cos ax}=x-\frac{1}{a}\tan \frac{ax}{2}+C$

$\int \frac{\cos ax\mathrm{d}x}{1-\cos ax}=-x-\frac{1}{a}\cot \frac{ax}{2}+C$

$\int \cos a_{1}x\cos a_{2}x\mathrm{d}x=\frac{\sin (a_2-a_1)x}{2(a_2-a_1)}+\frac{\sin (a_2+a_1)x}{2(a_2+a_1)}+C\text{(for }\left|a_1\right|\ne \left|a_2\right|\text{)}$

Integrands involving only tangent

$\int \tan ax\mathrm{d}x=-\frac{1}{a}\ln |\cos ax|+C=\frac{1}{a}\ln |\sec ax|+C$

$\int {\tan }^{2}x\mathrm{d}x=\tan x-x+C$

$\int {\tan }^{n}ax\mathrm{d}x=\frac{1}{a(n-1)}{\tan }^{n-1}ax-\int {\tan }^{n-2}ax\mathrm{d}x\text{(for }n\ne 1\text{)}$

$\int \frac{\mathrm{d}x}{q\tan ax+p}=\frac{1}{p^2+q^2}(px+\frac{q}{a}\ln |q\sin ax+p\cos ax|)+C\text{(for }{p}^2+q^2\ne 0\text{)}$

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

$\int \frac{\mathrm{d}x}{\tan ax-1}=-\frac{x}{2}+\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

$\int \frac{\tan ax\mathrm{d}x}{\tan ax+1}=\frac{x}{2}-\frac{1}{2a}\ln |\sin ax+\cos ax|+C$

$\int \frac{\tan ax\mathrm{d}x}{\tan ax-1}=\frac{x}{2}+\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

Integrands involving only secant

See Integral of the secant function.

$\int \sec ax\mathrm{d}x=\frac{1}{a}\ln |\sec ax+\tan ax|+C$

$\int {\sec }^{2}x\mathrm{d}x=\tan x+C$

$\int {\sec }^{3}xdx=\frac{1}{2}\sec x\tan x+\frac{1}{2}\ln |\sec x+\tan x|+C.$

$\int {\sec }^{n}ax\mathrm{d}x=\frac{{\sec }^{n-2}ax\tan ax}{a(n-1)}+\frac{n-2}{n-1}\int {\sec }^{n-2}ax\mathrm{d}x\text{ (for }n\ne 1\text{)}$

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

Integrands involving only cosecant

$\int \csc ax\mathrm{d}x=-\frac{1}{a}\ln |\csc ax+\cot ax|+C$

$\int {\csc }^{2}x\mathrm{d}x=-\cot x+C$

$\int {\csc }^{n}ax\mathrm{d}x=-\frac{{\csc }^{n-1}ax\cos ax}{a(n-1)}+\frac{n-2}{n-1}\int {\csc }^{n-2}ax\mathrm{d}x\text{ (for }n\ne 1\text{)}$

$\int \frac{\mathrm{d}x}{\csc x+1}=x-\frac{2\sin \frac{x}{2}}{\cos \frac{x}{2}+\sin \frac{x}{2}}+C$

$\int \frac{\mathrm{d}x}{\csc x-1}=\frac{2\sin \frac{x}{2}}{\cos \frac{x}{2}-\sin \frac{x}{2}}-x+C$

Integrands involving only cotangent

$\int \cot ax\mathrm{d}x=\frac{1}{a}\ln |\sin ax|+C$

$\int {\cot }^{n}ax\mathrm{d}x=-\frac{1}{a(n-1)}{\cot }^{n-1}ax-\int {\cot }^{n-2}ax\mathrm{d}x\text{(for }n\ne 1\text{)}$

$\int \frac{\mathrm{d}x}{1+\cot ax}=\int \frac{\tan ax\mathrm{d}x}{\tan ax+1}$

$\int \frac{\mathrm{d}x}{1-\cot ax}=\int \frac{\tan ax\mathrm{d}x}{\tan ax-1}$

Integrands involving both sine and cosine

$\int \frac{\mathrm{d}x}{\cos ax\pm \sin ax}=\frac{1}{a\sqrt{2}}\ln |\tan (\frac{ax}{2}\pm \frac{\pi }{8})|+C$

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

$\int \frac{\mathrm{d}x}{(\cos x+\sin x{)}^n}=\frac{1}{n-1}(\frac{\sin x-\cos x}{(\cos x+\sin x{)}^{n-1}}-2(n-2)\int \frac{\mathrm{d}x}{(\cos x+\sin x{)}^{n-2}})$

$\int \frac{\cos ax\mathrm{d}x}{\cos ax+\sin ax}=\frac{x}{2}+\frac{1}{2a}\ln |\sin ax+\cos ax|+C$

$\int \frac{\cos ax\mathrm{d}x}{\cos ax-\sin ax}=\frac{x}{2}-\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

$\int \frac{\sin ax\mathrm{d}x}{\cos ax+\sin ax}=\frac{x}{2}-\frac{1}{2a}\ln |\sin ax+\cos ax|+C$

$\int \frac{\sin ax\mathrm{d}x}{\cos ax-\sin ax}=-\frac{x}{2}-\frac{1}{2a}\ln |\sin ax-\cos ax|+C$

$\int \frac{\cos ax\mathrm{d}x}{\sin ax(1+\cos ax)}=-\frac{1}{4a}{\tan }^2\frac{ax}{2}+\frac{1}{2a}\ln |\tan \frac{ax}{2}|+C$

$\int \frac{\cos ax\mathrm{d}x}{\sin ax(1-\cos ax)}=-\frac{1}{4a}{\cot }^2\frac{ax}{2}-\frac{1}{2a}\ln |\tan \frac{ax}{2}|+C$

$\int \frac{\sin ax\mathrm{d}x}{\cos ax(1+\sin ax)}=\frac{1}{4a}{\cot }^2(\frac{ax}{2}+\frac{\pi }{4})+\frac{1}{2a}\ln |\tan (\frac{ax}{2}+\frac{\pi }{4})|+C$

$\int \frac{\sin ax\mathrm{d}x}{\cos ax(1-\sin ax)}=\frac{1}{4a}{\tan }^2(\frac{ax}{2}+\frac{\pi }{4})-\frac{1}{2a}\ln |\tan (\frac{ax}{2}+\frac{\pi }{4})|+C$

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

$\int \sin a_{1}x\cos a_{2}x\mathrm{d}x=-\frac{\cos ((a_1-a_2)x)}{2(a_1-a_2)}-\frac{\cos ((a_1+a_2)x)}{2(a_1+a_2)}+C\text{(for }\left|a_1\right|\ne \left|a_2\right|\text{)}$

$\int {\sin }^{n}ax\cos ax\mathrm{d}x=\frac{1}{a(n+1)}{\sin }^{}ax+C\text{(for }n\ne -1\text{)}$

$\int \sin ax{\cos }^{}ax\mathrm{d}x=-\frac{1}{a(n+1)}{\cos }^{}ax+C\text{(for }n\ne -1\text{)}$

$\int {\sin }^{n}ax{\cos }^{m}ax\mathrm{d}x=-\frac{{\sin }^{n-1}ax{\cos }^{m+1}ax}{a(n+m)}+\frac{n-1}{n+m}\int {\sin }^{n-2}ax{\cos }^{m}ax\mathrm{d}x\text{(for }m,n>0\text{)}$

also: $\int {\sin }^{n}ax{\cos }^{m}ax\mathrm{d}x=\frac{{\sin }^{n+1}ax{\cos }^{m-1}ax}{a(n+m)}+\frac{m-1}{n+m}\int {\sin }^{n}ax{\cos }^{m-2}ax\mathrm{d}x\text{(for }m,n>0\text{)}$

$\int \frac{\mathrm{d}x}{\sin ax\cos ax}=\frac{1}{a}\ln |\tan ax|+C$

$\int \frac{\mathrm{d}x}{\sin ax{\cos }^{}ax}=\frac{1}{a(n-1){\cos }^{n-1}ax}+\int \frac{\mathrm{d}x}{\sin ax{\cos }^{}ax}\text{(for }n\ne 1\text{)}$

$\int \frac{\mathrm{d}x}{{\sin }^{n}ax\cos ax}=-\frac{1}{a(n-1){\sin }^{n-1}ax}+\int \frac{\mathrm{d}x}{{\sin }^{n-2}ax\cos ax}\text{(for }n\ne 1\text{)}$

$\int \frac{\sin ax\mathrm{d}x}{{\cos }^{n}ax}=\frac{1}{a(n-1){\cos }^{n-1}ax}+C\text{(for }n\ne 1\text{)}$

$\int \frac{{\sin }^{2}ax\mathrm{d}x}{\cos ax}=-\frac{1}{a}\sin ax+\frac{1}{a}\ln |\tan (\frac{\pi }{4}+\frac{ax}{2})|+C$

$\int \frac{{\sin }^{2}ax\mathrm{d}x}{{\cos }^{n}ax}=\frac{\sin ax}{a(n-1){\cos }^{n-1}ax}-\frac{1}{n-1}\int \frac{\mathrm{d}x}{{\cos }^{n-2}ax}\text{(for }n\ne 1\text{)}$

$\int \frac{{\sin }^{n}ax\mathrm{d}x}{\cos ax}=-\frac{{\sin }^{n-1}ax}{a(n-1)}+\int \frac{{\sin }^{n-2}ax\mathrm{d}x}{\cos ax}\text{(for }n\ne 1\text{)}$

$\int \frac{{\sin }^{n}ax\mathrm{d}x}{{\cos }^{m}ax}=\frac{{\sin }^{n+1}ax}{a(m-1){\cos }^{m-1}ax}-\frac{n-m+2}{m-1}\int \frac{{\sin }^{n}ax\mathrm{d}x}{{\cos }^{m-2}ax}\text{(for }m\ne 1\text{)}$

also: $\int \frac{{\sin }^{n}ax\mathrm{d}x}{{\cos }^{m}ax}=-\frac{{\sin }^{n-1}ax}{a(n-m){\cos }^{m-1}ax}+\frac{n-1}{n-m}\int \frac{{\sin }^{n-2}ax\mathrm{d}x}{{\cos }^{m}ax}\text{(for }m\ne n\text{)}$

also: $\int \frac{{\sin }^{n}ax\mathrm{d}x}{{\cos }^{m}ax}=\frac{{\sin }^{n-1}ax}{a(m-1){\cos }^{m-1}ax}-\frac{n-1}{m-1}\int \frac{{\sin }^{n-2}ax\mathrm{d}x}{{\cos }^{m-2}ax}\text{(for }m\ne 1\text{)}$

$\int \frac{\cos ax\mathrm{d}x}{{\sin }^{n}ax}=-\frac{1}{a(n-1){\sin }^{n-1}ax}+C\text{(for }n\ne 1\text{)}$

$\int \frac{{\cos }^{2}ax\mathrm{d}x}{\sin ax}=\frac{1}{a}(\cos ax+\ln |\tan \frac{ax}{2}|)+C$

$\int \frac{{\cos }^{2}ax\mathrm{d}x}{{\sin }^{n}ax}=-\frac{1}{n-1}(\frac{\cos ax}{a{\sin }^{n-1}ax)}+\int \frac{\mathrm{d}x}{{\sin }^{n-2}ax})\text{(for }n\ne 1\text{)}$

$\int \frac{{\cos }^{n}ax\mathrm{d}x}{{\sin }^{m}ax}=-\frac{{\cos }^{n+1}ax}{a(m-1){\sin }^{m-1}ax}-\frac{n-m+2}{m-1}\int \frac{{\cos }^{n}ax\mathrm{d}x}{{\sin }^{m-2}ax}\text{(for }m\ne 1\text{)}$

also: $\int \frac{{\cos }^{n}ax\mathrm{d}x}{{\sin }^{m}ax}=\frac{{\cos }^{n-1}ax}{a(n-m){\sin }^{m-1}ax}+\frac{n-1}{n-m}\int \frac{{\cos }^{n-2}ax\mathrm{d}x}{{\sin }^{m}ax}\text{(for }m\ne n\text{)}$

also: $\int \frac{{\cos }^{n}ax\mathrm{d}x}{{\sin }^{m}ax}=-\frac{{\cos }^{n-1}ax}{a(m-1){\sin }^{m-1}ax}-\frac{n-1}{m-1}\int \frac{{\cos }^{n-2}ax\mathrm{d}x}{{\sin }^{m-2}ax}\text{(for }m\ne 1\text{)}$

Integrands involving both sine and tangent

$\int \sin ax\tan ax\mathrm{d}x=\frac{1}{a}(\ln |\sec ax+\tan ax|-\sin ax)+C$

$\int \frac{{\tan }^{n}ax\mathrm{d}x}{{\sin }^{2}ax}=\frac{1}{a(n-1)}{\tan }^{n-1}(ax)+C\text{(for }n\ne 1\text{)}$

Integrands involving both cosine and tangent

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

Integrals containing both sine and cotangent

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

Integrands involving both cosine and cotangent

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

Integrands involving both secant and tangent

$\int \sec x\tan xdx=\sec x+C$

Integrals with symmetric limits

$\underset{-c}{\overset{c}{\int }}\sin x\mathrm{d}x=0$

$\underset{-c}{\overset{c}{\int }}\cos x\mathrm{d}x=2\underset{0}{\overset{c}{\int }}\cos x\mathrm{d}x=2\underset{-c}{\overset{0}{\int }}\cos x\mathrm{d}x=2\sin c$

$\underset{-c}{\overset{c}{\int }}\tan x\mathrm{d}x=0$

$\underset{-\frac{a}{2}}{\overset{\frac{a}{2}}{\int }}{x}^2{\cos }^2\frac{n\pi x}{a}\mathrm{d}x=\frac{a^3(n^2{\pi }^2-6)}{24n^2{\pi }^2}\text{(for }n=1,3,5...\text{)}$

$\underset{\frac{-a}{2}}{\overset{\frac{a}{2}}{\int }}{x}^2{\sin }^2\frac{n\pi x}{a}\mathrm{d}x=\frac{a^3(n^2{\pi }^2-6(-1{)}^n)}{24n^2{\pi }^2}=\frac{a^3}{24}(1-6\frac{(-1{)}^n}{n^2{\pi }^2})\text{(for }n=1,2,3,...\text{)}$

Integral over a full circle

$\underset{0}{\overset{2\pi }{\int }}{\sin }^{2m+1}xco{s}^{2n+1}x\mathrm{d}x=0\{n,m\}\in \mathbb{\mathbb{Z}}$

References

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