List of integrals of trigonometric functions
|
|
The following is a list of integrals (antiderivative functions) of trigonometric functions. For a complete list of Integral functions, please see table of integrals and list of integrals. See also: trigonometric integral
The constant c is assumed to be nonzero.
| Contents |
Integrals of trigonometric functions containing only sin
- <math>\int\sin cx\;dx = -\frac{1}{c}\cos cx\,\!<math>
- <math>\int\sin^n cx\;dx = -\frac{\sin^{n-1} cx\cos cx}{nc} + \frac{n-1}{n}\int\sin^{n-2} cx\;dx \qquad\mbox{(for }n>0\mbox{)}\,\!<math>
- <math>\int\sqrt{1 - \sin{x}}\,dx = \int\sqrt{\operatorname{cvs}{x}}\,dx = 2 \frac{\cos{\frac{x}{2}} + \sin{\frac{x}{2}}}{\cos{\frac{x}{2}} - \sin{\frac{x}{2}}} \sqrt{\operatorname{cvs}{x}} <math>
where cvs{x} is the Coversine function
- <math>\int x\sin cx\;dx = \frac{\sin cx}{c^2}-\frac{x\cos cx}{c}\,\!<math>
- <math>\int x^n\sin cx\;dx = -\frac{x^n}{c}\cos cx+\frac{n}{c}\int x^{n-1}\cos cx\;dx \qquad\mbox{(for }n>0\mbox{)}\,\!<math>
- <math>\int\frac{\sin cx}{x} dx = \sum_{i=0}^\infty (-1)^i\frac{(cx)^{2i+1}}{(2i+1)\cdot (2i+1)!}\,\!<math>
- <math>\int\frac{\sin cx}{x^n} dx = -\frac{\sin cx}{(n-1)x^{n-1}} + \frac{c}{n-1}\int\frac{\cos cx}{x^{n-1}} dx\,\!<math>
- <math>\int\frac{dx}{\sin cx} = \frac{1}{c}\ln \left|\tan\frac{cx}{2}\right|<math>
- <math>\int\frac{dx}{\sin^n cx} = \frac{\cos cx}{c(1-n) \sin^{n-1} cx}+\frac{n-2}{n-1}\int\frac{dx}{\sin^{n-2}cx} \qquad\mbox{(for }n>1\mbox{)}\,\!<math>
- <math>\int\frac{dx}{1\pm\sin cx} = \frac{1}{c}\tan\left(\frac{cx}{2}\mp\frac{\pi}{4}\right)<math>
- <math>\int\frac{x\;dx}{1+\sin cx} = \frac{x}{c}\tan\left(\frac{cx}{2} - \frac{\pi}{4}\right)+\frac{2}{c^2}\ln\left|\cos\left(\frac{cx}{2}-\frac{\pi}{4}\right)\right|<math>
- <math>\int\frac{x\;dx}{1-\sin cx} = \frac{x}{c}\cot\left(\frac{\pi}{4} - \frac{cx}{2}\right)+\frac{2}{c^2}\ln\left|\sin\left(\frac{\pi}{4}-\frac{cx}{2}\right)\right|<math>
- <math>\int\frac{\sin cx\;dx}{1\pm\sin cx} = \pm x+\frac{1}{c}\tan\left(\frac{\pi}{4}\mp\frac{cx}{2}\right)<math>
- <math>\int\sin c_1x\sin c_2x\;dx = \frac{\sin(c_1-c_2)x}{2(c_1-c_2)}-\frac{\sin(c_1+c_2)x}{2(c_1+c_2)} \qquad\mbox{(for }|c_1|\neq|c_2|\mbox{)}\,\!<math>
Integrals of trigonometric functions containing only cos
- <math>\int\cos cx\;dx = \frac{1}{c}\sin cx\,\!<math>
- <math>\int\cos^n cx\;dx = -\frac{\cos^{n-1} cx\sin cx}{nc} + \frac{n-1}{n}\int\cos^{n-2} cx\;dx \qquad\mbox{(for }n>0\mbox{)}\,\!<math>
- <math>\int x\cos cx\;dx = \frac{\cos cx}{c^2} + \frac{x\sin cx}{c}\,\!<math>
- <math>\int x^n\cos cx\;dx = \frac{x^n\sin cx}{c} - \frac{n}{c}\int x^{n-1}\sin cx\;dx\,\!<math>
- <math>\int\frac{\cos cx}{x} dx = \ln|cx|+\sum_{i=1}^\infty (-1)^i\frac{(cx)^{2i}}{2i\cdot(2i)!}\,\!<math>
- <math>\int\frac{\cos cx}{x^n} dx = -\frac{\cos cx}{(n-1)x^{n-1}}-\frac{c}{n-1}\int\frac{\sin cx}{x^{n-1}} dx \qquad\mbox{(for }n\neq 1\mbox{)}\,\!<math>
- <math>\int\frac{dx}{\cos cx} = \frac{1}{c}\ln\left|\tan\left(\frac{cx}{2}+\frac{\pi}{4}\right)\right|<math>
- <math>\int\frac{dx}{\cos^n cx} = \frac{\sin cx}{c(n-1) cos^{n-1} cx} + \frac{n-2}{n-1}\int\frac{dx}{\cos^{n-2} cx} \qquad\mbox{(for }n>1\mbox{)}\,\!<math>
- <math>\int\frac{dx}{1+\cos cx} = \frac{1}{c}\tan\frac{cx}{2}\,\!<math>
- <math>\int\frac{dx}{1-\cos cx} = -\frac{1}{c}\cot\frac{cx}{2}\,\!<math>
- <math>\int\frac{x\;dx}{1+\cos cx} = \frac{x}{c}\tan{cx}{2} + \frac{2}{c^2}\ln\left|\cos\frac{cx}{2}\right|<math>
- <math>\int\frac{x\;dx}{1-\cos cx} = -\frac{x}{x}\cot{cx}{2}+\frac{2}{c^2}\ln\left|\sin\frac{cx}{2}\right|<math>
- <math>\int\frac{\cos cx\;dx}{1+\cos cx} = x - \frac{1}{c}\tan\frac{cx}{2}\,\!<math>
- <math>\int\frac{\cos cx\;dx}{1-\cos cx} = -x-\frac{1}{c}\cot\frac{cx}{2}\,\!<math>
- <math>\int\cos c_1x\cos c_2x\;dx = \frac{\sin(c_1-c_2)x}{2(c_1-c_2)}+\frac{\sin(c_1+c_2)x}{2(c_1+c_2)} \qquad\mbox{(for }|c_1|\neq|c_2|\mbox{)}\,\!<math>
Integrals of trigonometric functions containing only tan
- <math>\int\tan cx\;dx = -\frac{1}{c}\ln|\cos cx|\,\!<math>
- <math>\int\tan^n cx\;dx = \frac{1}{c(n-1)}\tan^{n-1} cx-\int\tan^{n-2} cx\;dx \qquad\mbox{(for }n\neq 1\mbox{)}\,\!<math>
- <math>\int\frac{dx}{\tan cx + 1} = \frac{x}{2} + \frac{1}{2c}\ln|\sin cx + \cos cx|\,\!<math>
- <math>\int\frac{dx}{\tan cx - 1} = -\frac{x}{2} + \frac{1}{2c}\ln|\sin cx - \cos cx|\,\!<math>
- <math>\int\frac{\tan cx\;dx}{\tan cx + 1} = \frac{x}{2} - \frac{1}{2c}\ln|\sin cx + \cos cx|\,\!<math>
- <math>\int\frac{\tan cx\;dx}{\tan cx - 1} = \frac{x}{2} + \frac{1}{2c}\ln|\sin cx - \cos cx|\,\!<math>
Integrals of trigonometric functions containing only sec
- <math>\int \sec{cx} \, dx = \frac{1}{c}\ln{\left| \sec{cx} + \tan{cx}\right|}<math>
- <math>\int \sec^n{cx} \, dx = \frac{\sec^{n-1}{cx} \sin {cx}}{c(n-1)} \,+\, \frac{n-2}{n-1}\int \sec^{n-2}{cx} \, dx \qquad \mbox{ (for }n \ne 1\mbox{)}\,\!<math>
- <math>\int \frac{dx}{\sec{x} + 1} = x - \tan{\frac{x}{2}}<math>
Integrals of trigonometric functions containing only csc
- <math>\int \csc{cx} \, dx = -\frac{1}{c}\ln{\left| \csc{cx} + \cot{cx}\right|}<math>
- <math>\int \csc^n{cx} \, dx = -\frac{\csc^{n-1}{cx} \cos{cx}}{c(n-1)} \,+\, \frac{n-2}{n-1}\int \csc^{n-2}{cx} \, dx \qquad \mbox{ (for }n \ne 1\mbox{)}\,\!<math>
Integrals of trigonometric functions containing only cot
- <math>\int\cot cx\;dx = \frac{1}{c}\ln|\sin cx|\,\!<math>
- <math>\int\cot^n cx\;dx = -\frac{1}{c(n-1)}\cot^{n-1} cx - \int\cot^{n-2} cx\;dx \qquad\mbox{(for }n\neq 1\mbox{)}\,\!<math>
- <math>\int\frac{dx}{1 + \cot cx} = \int\frac{\tan cx\;dx}{\tan cx+1}\,\!<math>
- <math>\int\frac{dx}{1 - \cot cx} = \int\frac{\tan cx\;dx}{\tan cx-1}\,\!<math>
Integrals of trigonometric functions containing both sin and cos
- <math>\int\frac{dx}{\cos cx\pm\sin cx} = \frac{1}{c\sqrt{2}}\ln\left|\tan\left(\frac{cx}{2}\pm\frac{\pi}{8}\right)\right|<math>
- <math>\int\frac{dx}{(\cos cx\pm\sin cx)^2} = \frac{1}{2c}\tan\left(cx\mp\frac{\pi}{4}\right)<math>
- <math>\int\frac{dx}{(\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{dx}{(\cos x + \sin x)^{n-2}} \right)<math>
- <math>\int\frac{\cos cx\;dx}{\cos cx + \sin cx} = \frac{x}{2} + \frac{1}{2c}\ln\left|\sin cx + \cos cx\right|<math>
- <math>\int\frac{\cos cx\;dx}{\cos cx - \sin cx} = \frac{x}{2} - \frac{1}{2c}\ln\left|\sin cx - \cos cx\right|<math>
- <math>\int\frac{\sin cx\;dx}{\cos cx + \sin cx} = \frac{x}{2} - \frac{1}{2c}\ln\left|\sin cx + \cos cx\right|<math>
- <math>\int\frac{\sin cx\;dx}{\cos cx - \sin cx} = -\frac{x}{2} - \frac{1}{2c}\ln\left|\sin cx - \cos cx\right|<math>
- <math>\int\frac{\cos cx\;dx}{\sin cx(1+\cos cx)} = -\frac{1}{4c}\tan^2\frac{cx}{2}+\frac{1}{2c}\ln\left|\tan\frac{cx}{2}\right|<math>
- <math>\int\frac{\cos cx\;dx}{\sin cx(1+-\cos cx)} = -\frac{1}{4c}\cot^2\frac{cx}{2}-\frac{1}{2c}\ln\left|\tan\frac{cx}{2}\right|<math>
- <math>\int\frac{\sin cx\;dx}{\cos cx(1+\sin cx)} = \frac{1}{4c}\cot^2\left(\frac{cx}{2}+\frac{\pi}{4}\right)+\frac{1}{2c}\ln\left|\tan\left(\frac{cx}{2}+\frac{\pi}{4}\right)\right|<math>
- <math>\int\frac{\sin cx\;dx}{\cos cx(1-\sin cx)} = \frac{1}{4c}\tan^2\left(\frac{cx}{2}+\frac{\pi}{4}\right)-\frac{1}{2c}\ln\left|\tan\left(\frac{cx}{2}+\frac{\pi}{4}\right)\right|<math>
- <math>\int\sin cx\cos cx\;dx = \frac{1}{2c}\sin^2 cx\,\!<math>
- <math>\int\sin c_1x\cos c_2x\;dx = -\frac{\cos(c_1+c_2)x}{2(c_1+c_2)}-\frac{\cos(c_1-c_2)x}{2(c_1-c_2)} \qquad\mbox{(for }|c_1|\neq|c_2|\mbox{)}\,\!<math>
- <math>\int\sin^n cx\cos cx\;dx = \frac{1}{c(n+1)}\sin^{n+1} cx \qquad\mbox{(for }n\neq 1\mbox{)}\,\!<math>
- <math>\int\sin cx\cos^n cx\;dx = -\frac{1}{c(n+1)}\cos^{n+1} cx \qquad\mbox{(for }n\neq 1\mbox{)}\,\!<math>
- <math>\int\sin^n cx\cos^m cx\;dx = -\frac{\sin^{n-1} cx\cos^{m+1} cx}{c(n+m)}+\frac{n-1}{n+m}\int\sin^{n-2} cx\cos^m cx\;dx \qquad\mbox{(for }m,n>0\mbox{)}\,\!<math>
- also: <math>\int\sin^n cx\cos^m cx\;dx = \frac{\sin^{n+1} cx\cos^{m-1} cx}{c(n+m)} + \frac{m-1}{n+m}\int\sin^n cx\cos^{m-2} cx\;dx \qquad\mbox{(for }m,n>0\mbox{)}\,\!<math>
- <math>\int\frac{dx}{\sin cx\cos cx} = \frac{1}{c}\ln\left|\tan cx\right|<math>
- <math>\int\frac{dx}{\sin cx\cos^n cx} = \frac{1}{c(n-1)\cos^{n-1} cx}+\int\frac{dx}{\sin cx\cos^{n-2} cx} \qquad\mbox{(for }n\neq 1\mbox{)}\,\!<math>
- <math>\int\frac{dx}{\sin^n cx\cos cx} = -\frac{1}{c(n-1)\sin^{n-1} cx}+\int\frac{dx}{\sin^{n-2} cx\cos cx} \qquad\mbox{(for }n\neq 1\mbox{)}\,\!<math>
- <math>\int\frac{\sin cx\;dx}{\cos^n cx} = \frac{1}{c(n-1)\cos^{n-1} cx} \qquad\mbox{(for }n\neq 1\mbox{)}\,\!<math>
- <math>\int\frac{\sin^2 cx\;dx}{\cos cx} = -\frac{1}{c}\sin cx+\frac{1}{c}\ln\left|\tan\left(\frac{\pi}{4}+\frac{cx}{2}\right)\right|<math>
- <math>\int\frac{\sin^2 cx\;dx}{\cos^n cx} = \frac{\sin cx}{c(n-1)\cos^{n-1}cx}-\frac{1}{n-1}\int\frac{dx}{\cos^{n-2}cx} \qquad\mbox{(for }n\neq 1\mbox{)}\,\!<math>
- <math>\int\frac{\sin^n cx\;dx}{\cos cx} = -\frac{\sin^{n-1} cx}{c(n-1)} + \int\frac{\sin^{n-2} cx\;dx}{\cos cx} \qquad\mbox{(for }n\neq 1\mbox{)}\,\!<math>
- <math>\int\frac{\sin^n cx\;dx}{\cos^m cx} = \frac{\sin^{n+1} cx}{c(m-1)\cos^{m-1} cx}-\frac{n-m+2}{m-1}\int\frac{\sin^n cx\;dx}{\cos^{m-2} cx} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!<math>
- also: <math>\int\frac{\sin^n cx\;dx}{\cos^m cx} = -\frac{\sin^{n-1} cx}{c(n-m)\cos^{m-1} cx}+\frac{n-1}{n-m}\int\frac{\sin^{n-2} cx\;dx}{\cos^m cx} \qquad\mbox{(for }m\neq n\mbox{)}\,\!<math>
- also: <math>\int\frac{\sin^n cx\;dx}{\cos^m cx} = \frac{\sin^{n-1} cx}{c(m-1)\cos^{m-1} cx}-\frac{n-1}{n-1}\int\frac{\sin^{n-1} cx\;dx}{\cos^{m-2} cx} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!<math>
- <math>\int\frac{\cos cx\;dx}{\sin^n cx} = -\frac{1}{c(n-1)\sin^{n-1} cx} \qquad\mbox{(for }n\neq 1\mbox{)}\,\!<math>
- <math>\int\frac{\cos^2 cx\;dx}{\sin cx} = \frac{1}{c}\left(\cos cx+\ln\left|\tan\frac{cx}{2}\right|\right)<math>
- <math>\int\frac{\cos^2 cx\;dx}{\sin^n cx} = -\frac{1}{n-1}\left(\frac{\cos cx}{c\sin^{n-1} cx)}+\int\frac{dx}{\sin^{n-2} cx}\right) \qquad\mbox{(for }n\neq 1\mbox{)}<math>
- <math>\int\frac{\cos^n cx\;dx}{\sin^m cx} = -\frac{\cos^{n+1} cx}{c(m-1)\sin^{m-1} cx} - \frac{n-m-2}{m-1}\int\frac{cos^n cx\;dx}{\sin^{m-2} cx} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!<math>
- also: <math>\int\frac{\cos^n cx\;dx}{\sin^m cx} = \frac{\cos^{n-1} cx}{c(n-m)\sin^{m-1} cx} + \frac{n-1}{n-m}\int\frac{cos^{n-2} cx\;dx}{\sin^m cx} \qquad\mbox{(for }m\neq n\mbox{)}\,\!<math>
- also: <math>\int\frac{\cos^n cx\;dx}{\sin^m cx} = -\frac{\cos^{n-1} cx}{c(m-1)\sin^{m-1} cx} - \frac{n-1}{m-1}\int\frac{cos^{n-2} cx\;dx}{\sin^{m-2} cx} \qquad\mbox{(for }m\neq 1\mbox{)}\,\!<math>
Integrals of trigonometric functions containing both sin and tan
- <math>\int \sin cx \tan cx\;dx = \frac{1}{c}(\ln|\sec cx + \tan cx| - \sin cx)\,\!<math>
- <math>\int\frac{\tan^n cx\;dx}{\sin^2 cx} = \frac{1}{c(n-1)}\tan^{n-1} (cx) \qquad\mbox{(for }n\neq 1\mbox{)}\,\!<math>
Integrals of trigonometric functions containing both cos and tan
- <math>\int\frac{\tan^n cx\;dx}{\cos^2 cx} = \frac{1}{c(n+1)}\tan^{n+1} cx \qquad\mbox{(for }n\neq -1\mbox{)}\,\!<math>
Integrals of trigonometric functions containing both sin and cot
- <math>\int\frac{\cot^n cx\;dx}{\sin^2 cx} = \frac{1}{c(n+1)}\cot^{n+1} cx \qquad\mbox{(for }n\neq -1\mbox{)}\,\!<math>
Integrals of trigonometric functions containing both cos and cot
- <math>\int\frac{\cot^n cx\;dx}{\cos^2 cx} = \frac{1}{c(1-n)}\tan^{1-n} cx \qquad\mbox{(for }n\neq 1\mbox{)}\,\!<math>
Integrals of trigonometric functions containing both tan and cot
- <math>\int \frac{\tan^m(cx)}{\cot^n(cx)}\;dx = \frac{1}{c(m+n-1)}\tan^{m+n-1}(cx) - \int \frac{\tan^{m-2}(cx)}{\cot^n(cx)}\;dx\qquad\mbox{(for }m + n \neq 1\mbox{)}\,\!<math>fr:Primitives de fonctions trigonométriques
it:Tavola degli integrali indefiniti di funzioni trigonometriche pl:Całki funkcji trygonometrycznych