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. See also trigonometric integral.

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%2Bc

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

Contents

Integrals involving only sine

\int\sin ax\;dx = -\frac{1}{a}\cos ax%2BC\,\!
\int\sin^2 {ax}\;dx = \frac{x}{2} - \frac{1}{4a} \sin 2ax %2BC= \frac{x}{2} - \frac{1}{2a} \sin ax\cos ax %2BC\!
\int x\sin^2 {ax}\;dx = \frac{x^2}{4} - \frac{x}{4a} \sin 2ax - \frac{1}{8a^2} \cos 2ax %2BC\!
\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 %2BC\!
\int\sin b_1x\sin b_2x\;dx = \frac{\sin((b_1-b_2)x)}{2(b_1-b_2)}-\frac{\sin((b_1%2Bb_2)x)}{2(b_1%2Bb_2)}%2BC \qquad\mbox{(for }|b_1|\neq|b_2|\mbox{)}\,\!
\int\sin^n {ax}\;dx = -\frac{\sin^{n-1} ax\cos ax}{na} %2B \frac{n-1}{n}\int\sin^{n-2} ax\;dx \qquad\mbox{(for }n>2\mbox{)}\,\!
\int\frac{dx}{\sin ax} = \frac{1}{a}\ln \left|\tan\frac{ax}{2}\right|%2BC
\int\frac{dx}{\sin^n ax} = \frac{\cos ax}{a(1-n) \sin^{n-1} ax}%2B\frac{n-2}{n-1}\int\frac{dx}{\sin^{n-2}ax} \qquad\mbox{(for }n>1\mbox{)}\,\!
\int x\sin ax\;dx = \frac{\sin ax}{a^2}-\frac{x\cos ax}{a}%2BC\,\!
\int x^n\sin ax\;dx = -\frac{x^n}{a}\cos ax%2B\frac{n}{a}\int x^{n-1}\cos ax\;dx = \sum_{k=0}^{2k\leq n} (-1)^{k%2B1} \frac{x^{n-2k}}{a^{1%2B2k}}\frac{n!}{(n-2k)!} \cos ax %2B\sum_{k=0}^{2k%2B1\leq n}(-1)^k \frac{x^{n-1-2k}}{a^{2%2B2k}}\frac{n!}{(n-2k-1)!} \sin ax \qquad\mbox{(for }n>0\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)}{24n^2\pi^2} \qquad\mbox{(for }n=2,4,6...\mbox{)}\,\!
\int\frac{\sin ax}{x} dx = \sum_{n=0}^\infty (-1)^n\frac{(ax)^{2n%2B1}}{(2n%2B1)\cdot (2n%2B1)!} %2BC\,\!
\int\frac{\sin ax}{x^n} dx = -\frac{\sin ax}{(n-1)x^{n-1}} %2B \frac{a}{n-1}\int\frac{\cos ax}{x^{n-1}} dx\,\!
\int\frac{dx}{1\pm\sin ax} = \frac{1}{a}\tan\left(\frac{ax}{2}\mp\frac{\pi}{4}\right)%2BC
\int\frac{x\;dx}{1%2B\sin ax} = \frac{x}{a}\tan\left(\frac{ax}{2} - \frac{\pi}{4}\right)%2B\frac{2}{a^2}\ln\left|\cos\left(\frac{ax}{2}-\frac{\pi}{4}\right)\right|%2BC
\int\frac{x\;dx}{1-\sin ax} = \frac{x}{a}\cot\left(\frac{\pi}{4} - \frac{ax}{2}\right)%2B\frac{2}{a^2}\ln\left|\sin\left(\frac{\pi}{4}-\frac{ax}{2}\right)\right|%2BC
\int\frac{\sin ax\;dx}{1\pm\sin ax} = \pm x%2B\frac{1}{a}\tan\left(\frac{\pi}{4}\mp\frac{ax}{2}\right)%2BC

Integrands involving only cosine

\int\cos ax\;dx = \frac{1}{a}\sin ax%2BC\,\!
\int\cos^2 {ax}\;dx = \frac{x}{2} %2B \frac{1}{4a} \sin 2ax %2BC = \frac{x}{2} %2B \frac{1}{2a} \sin ax\cos ax %2BC\!
\int\cos^n ax\;dx = \frac{\cos^{n-1} ax\sin ax}{na} %2B \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} %2B \frac{x\sin ax}{a}%2BC\,\!
\int x^2\cos^2 {ax}\;dx = \frac{x^3}{6} %2B \left( \frac {x^2}{4a} - \frac{1}{8a^3} \right) \sin 2ax %2B \frac{x}{4a^2} \cos 2ax %2BC\!
\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%2B1\leq n} (-1)^{k} \frac{x^{n-2k-1}}{a^{2%2B2k}}\frac{n!}{(n-2k-1)!} \cos ax %2B\sum_{k=0}^{2k\leq n}(-1)^{k} \frac{x^{n-2k}}{a^{1%2B2k}}\frac{n!}{(n-2k)!} \sin ax \!
\int\frac{\cos ax}{x} dx = \ln|ax|%2B\sum_{k=1}^\infty (-1)^k\frac{(ax)^{2k}}{2k\cdot(2k)!}%2BC\,\!
\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}%2B\frac{\pi}{4}\right)\right|%2BC
\int\frac{dx}{\cos^n ax} = \frac{\sin ax}{a(n-1) \cos^{n-1} ax} %2B \frac{n-2}{n-1}\int\frac{dx}{\cos^{n-2} ax} \qquad\mbox{(for }n>1\mbox{)}\,\!
\int\frac{dx}{1%2B\cos ax} = \frac{1}{a}\tan\frac{ax}{2}%2BC\,\!
\int\frac{dx}{1-\cos ax} = -\frac{1}{a}\cot\frac{ax}{2}%2BC\,\!
\int\frac{x\;dx}{1%2B\cos ax} = \frac{x}{a}\tan\frac{ax}{2} %2B \frac{2}{a^2}\ln\left|\cos\frac{ax}{2}\right|%2BC
\int\frac{x\;dx}{1-\cos ax} = -\frac{x}{a}\cot\frac{ax}{2}%2B\frac{2}{a^2}\ln\left|\sin\frac{ax}{2}\right|%2BC
\int\frac{\cos ax\;dx}{1%2B\cos ax} = x - \frac{1}{a}\tan\frac{ax}{2}%2BC\,\!
\int\frac{\cos ax\;dx}{1-\cos ax} = -x-\frac{1}{a}\cot\frac{ax}{2}%2BC\,\!
\int\cos a_1x\cos a_2x\;dx = \frac{\sin(a_1-a_2)x}{2(a_1-a_2)}%2B\frac{\sin(a_1%2Ba_2)x}{2(a_1%2Ba_2)}%2BC \qquad\mbox{(for }|a_1|\neq|a_2|\mbox{)}\,\!

Integrands involving only tangent

\int\tan ax\;dx = -\frac{1}{a}\ln|\cos ax|%2BC = \frac{1}{a}\ln|\sec ax|%2BC\,\!
\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 %2B p} = \frac{1}{p^2 %2B q^2}(px %2B \frac{q}{a}\ln|q\sin ax %2B p\cos ax|)%2BC \qquad\mbox{(for }p^2 %2B q^2\neq 0\mbox{)}\,\!
\int\frac{dx}{\tan ax} = \frac{1}{a}\ln|\sin ax|%2BC\,\!
\int\frac{dx}{\tan ax %2B 1} = \frac{x}{2} %2B \frac{1}{2a}\ln|\sin ax %2B \cos ax|%2BC\,\!
\int\frac{dx}{\tan ax - 1} = -\frac{x}{2} %2B \frac{1}{2a}\ln|\sin ax - \cos ax|%2BC\,\!
\int\frac{\tan ax\;dx}{\tan ax %2B 1} = \frac{x}{2} - \frac{1}{2a}\ln|\sin ax %2B \cos ax|%2BC\,\!
\int\frac{\tan ax\;dx}{\tan ax - 1} = \frac{x}{2} %2B \frac{1}{2a}\ln|\sin ax - \cos ax|%2BC\,\!

Integrands involving only secant

In the 17th century, the integral of the secant function was the subject of a well-known conjecture posed in the 1640s by Henry Bond. The problem was solved by Isaac Barrow.[1] It was originally for the purposes of cartography that this was needed. See Integral of the secant function.

\int \sec{ax} \, dx = \frac{1}{a}\ln{\left| \sec{ax} %2B \tan{ax}\right|}%2BC
\int \sec^2{x} \, dx = \tan{x}%2BC
\int \sec^n{ax} \, dx = \frac{\sec^{n-2}{ax} \tan {ax}}{a(n-1)} \,%2B\, \frac{n-2}{n-1}\int \sec^{n-2}{ax} \, dx \qquad \mbox{ (for }n \ne 1\mbox{)}\,\!
\int \sec^n{x} \, dx = \frac{\sec^{n-2}{x}\tan{x}}{n-1} \,%2B\, \frac{n-2}{n-1}\int \sec^{n-2}{x}\,dx[2]
\int \frac{dx}{\sec{x} %2B 1} = x - \tan{\frac{x}{2}}%2BC
\int \frac{dx}{\sec{x} - 1} = - x - \cot{\frac{x}{2}}%2BC

Integrands involving only cosecant

\int \csc{ax} \, dx = -\frac{1}{a}\ln{\left| \csc{ax}%2B\cot{ax}\right|}%2BC
\int \csc^2{x} \, dx = -\cot{x}%2BC
\int \csc^n{ax} \, dx = -\frac{\csc^{n-1}{ax} \cos{ax}}{a(n-1)} \,%2B\, \frac{n-2}{n-1}\int \csc^{n-2}{ax} \, dx \qquad \mbox{ (for }n \ne 1\mbox{)}\,\!
\int \frac{dx}{\csc{x} %2B 1} = x - \frac{2\sin{\frac{x}{2}}}{\cos{\frac{x}{2}}%2B\sin{\frac{x}{2}}}%2BC
\int \frac{dx}{\csc{x} - 1} = \frac{2\sin{\frac{x}{2}}}{\cos{\frac{x}{2}}-\sin{\frac{x}{2}}}-x%2BC

Integrands involving only cotangent

\int\cot ax\;dx = \frac{1}{a}\ln|\sin ax|%2BC\,\!
\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 %2B \cot ax} = \int\frac{\tan ax\;dx}{\tan ax%2B1}\,\!
\int\frac{dx}{1 - \cot ax} = \int\frac{\tan ax\;dx}{\tan ax-1}\,\!

Integrands involving both sine and cosine

\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|%2BC
\int\frac{dx}{(\cos ax\pm\sin ax)^2} = \frac{1}{2a}\tan\left(ax\mp\frac{\pi}{4}\right)%2BC
\int\frac{dx}{(\cos x %2B \sin x)^n} = \frac{1}{n-1}\left(\frac{\sin x - \cos x}{(\cos x %2B \sin x)^{n - 1}} - 2(n - 2)\int\frac{dx}{(\cos x %2B \sin x)^{n-2}} \right)
\int\frac{\cos ax\;dx}{\cos ax %2B \sin ax} = \frac{x}{2} %2B \frac{1}{2a}\ln\left|\sin ax %2B \cos ax\right|%2BC
\int\frac{\cos ax\;dx}{\cos ax - \sin ax} = \frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax - \cos ax\right|%2BC
\int\frac{\sin ax\;dx}{\cos ax %2B \sin ax} = \frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax %2B \cos ax\right|%2BC
\int\frac{\sin ax\;dx}{\cos ax - \sin ax} = -\frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax - \cos ax\right|%2BC
\int\frac{\cos ax\;dx}{\sin ax(1%2B\cos ax)} = -\frac{1}{4a}\tan^2\frac{ax}{2}%2B\frac{1}{2a}\ln\left|\tan\frac{ax}{2}\right|%2BC
\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|%2BC
\int\frac{\sin ax\;dx}{\cos ax(1%2B\sin ax)} = \frac{1}{4a}\cot^2\left(\frac{ax}{2}%2B\frac{\pi}{4}\right)%2B\frac{1}{2a}\ln\left|\tan\left(\frac{ax}{2}%2B\frac{\pi}{4}\right)\right|%2BC
\int\frac{\sin ax\;dx}{\cos ax(1-\sin ax)} = \frac{1}{4a}\tan^2\left(\frac{ax}{2}%2B\frac{\pi}{4}\right)-\frac{1}{2a}\ln\left|\tan\left(\frac{ax}{2}%2B\frac{\pi}{4}\right)\right|%2BC
\int\sin ax\cos ax\;dx = -\frac{1}{2a}\cos^2 ax %2BC\,\!
\int\sin a_1x\cos a_2x\;dx = -\frac{\cos((a_1-a_2)x)}{2(a_1-a_2)} -\frac{\cos((a_1%2Ba_2)x)}{2(a_1%2Ba_2)} %2BC\qquad\mbox{(for }|a_1|\neq|a_2|\mbox{)}\,\!
\int\sin^n ax\cos ax\;dx = \frac{1}{a(n%2B1)}\sin^{n%2B1} ax %2BC\qquad\mbox{(for }n\neq -1\mbox{)}\,\!
\int\sin ax\cos^n ax\;dx = -\frac{1}{a(n%2B1)}\cos^{n%2B1} ax %2BC\qquad\mbox{(for }n\neq -1\mbox{)}\,\!
\int\sin^n ax\cos^m ax\;dx = -\frac{\sin^{n-1} ax\cos^{m%2B1} ax}{a(n%2Bm)}%2B\frac{n-1}{n%2Bm}\int\sin^{n-2} ax\cos^m ax\;dx \qquad\mbox{(for }m,n>0\mbox{)}\,\!
also: \int\sin^n ax\cos^m ax\;dx = \frac{\sin^{n%2B1} ax\cos^{m-1} ax}{a(n%2Bm)} %2B \frac{m-1}{n%2Bm}\int\sin^n ax\cos^{m-2} ax\;dx \qquad\mbox{(for }m,n>0\mbox{)}\,\!
\int\frac{dx}{\sin ax\cos ax} = \frac{1}{a}\ln\left|\tan ax\right|%2BC
\int\frac{dx}{\sin ax\cos^n ax} = \frac{1}{a(n-1)\cos^{n-1} ax}%2B\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}%2B\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} %2BC\qquad\mbox{(for }n\neq 1\mbox{)}\,\!
\int\frac{\sin^2 ax\;dx}{\cos ax} = -\frac{1}{a}\sin ax%2B\frac{1}{a}\ln\left|\tan\left(\frac{\pi}{4}%2B\frac{ax}{2}\right)\right|%2BC
\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{)}\,\!
\int\frac{\sin^n ax\;dx}{\cos ax} = -\frac{\sin^{n-1} ax}{a(n-1)} %2B \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} = \frac{\sin^{n%2B1} ax}{a(m-1)\cos^{m-1} ax}-\frac{n-m%2B2}{m-1}\int\frac{\sin^n ax\;dx}{\cos^{m-2} ax} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!
also: \int\frac{\sin^n ax\;dx}{\cos^m ax} = -\frac{\sin^{n-1} ax}{a(n-m)\cos^{m-1} ax}%2B\frac{n-1}{n-m}\int\frac{\sin^{n-2} ax\;dx}{\cos^m ax} \qquad\mbox{(for }m\neq n\mbox{)}\,\!
also: \int\frac{\sin^n ax\;dx}{\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\;dx}{\cos^{m-2} ax} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!
\int\frac{\cos ax\;dx}{\sin^n ax} = -\frac{1}{a(n-1)\sin^{n-1} ax} %2BC\qquad\mbox{(for }n\neq 1\mbox{)}\,\!
\int\frac{\cos^2 ax\;dx}{\sin ax} = \frac{1}{a}\left(\cos ax%2B\ln\left|\tan\frac{ax}{2}\right|\right) %2BC
\int\frac{\cos^2 ax\;dx}{\sin^n ax} = -\frac{1}{n-1}\left(\frac{\cos ax}{a\sin^{n-1} ax)}%2B\int\frac{dx}{\sin^{n-2} ax}\right) \qquad\mbox{(for }n\neq 1\mbox{)}
\int\frac{\cos^n ax\;dx}{\sin^m ax} = -\frac{\cos^{n%2B1} ax}{a(m-1)\sin^{m-1} ax} - \frac{n-m-2}{m-1}\int\frac{\cos^n ax\;dx}{\sin^{m-2} ax} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!
also: \int\frac{\cos^n ax\;dx}{\sin^m ax} = \frac{\cos^{n-1} ax}{a(n-m)\sin^{m-1} ax} %2B \frac{n-1}{n-m}\int\frac{\cos^{n-2} ax\;dx}{\sin^m ax} \qquad\mbox{(for }m\neq n\mbox{)}\,\!
also: \int\frac{\cos^n ax\;dx}{\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\;dx}{\sin^{m-2} ax} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!

Integrands involving both sine and tangent

\int \sin ax \tan ax\;dx = \frac{1}{a}(\ln|\sec ax %2B \tan ax| - \sin ax)%2BC\,\!
\int\frac{\tan^n ax\;dx}{\sin^2 ax} = \frac{1}{a(n-1)}\tan^{n-1} (ax) %2BC\qquad\mbox{(for }n\neq 1\mbox{)}\,\!

Integrands involving both cosine and tangent

\int\frac{\tan^n ax\;dx}{\cos^2 ax} = \frac{1}{a(n%2B1)}\tan^{n%2B1} ax %2BC\qquad\mbox{(for }n\neq -1\mbox{)}\,\!

Integrals containing both sine and cotangent

\int\frac{\cot^n ax\;dx}{\sin^2 ax} = \frac{1}{a(n%2B1)}\cot^{n%2B1} ax %2BC\qquad\mbox{(for }n\neq -1\mbox{)}\,\!

Integrands involving both cosine and cotangent

\int\frac{\cot^n ax\;dx}{\cos^2 ax} = \frac{1}{a(1-n)}\tan^{1-n} ax %2BC\qquad\mbox{(for }n\neq 1\mbox{)}\,\!

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{)}\,\!

References

  1. ^ V. Frederick Rickey and Philip M. Tuchinsky, "An Application of Geography to Mathematics: History of the Integral of the Secant", Mathematics Magazine, volume 53, number 3, May 2980, pages 162–166
  2. ^ Stewart, James. Calculus: Early Transcendentals, 6th Edition. Thomson: 2008