List of integrals of trigonometric functions

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,

$\int a\cos nx\,dx = \frac\sin nx+C$

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

Integrands involving only

sine

* $\int\sin ax\,dx = -\frac\cos ax+C$
* $\int\sin^2 \,dx = \frac - \frac \sin 2ax +C= \frac - \frac \sin ax\cos ax +C$
* $\int\sin^3 \,dx = \frac - \frac +C$
* $\int x\sin^2 \,dx = \frac - \frac \sin 2ax - \frac \cos 2ax +C$
* $\int x^2\sin^2 \,dx = \frac - \left( \frac - \frac \right) \sin 2ax - \frac \cos 2ax +C$
* $\int x\sin ax\,dx = \frac-\frac+C$
* $\int(\sin b_1x)(\sin b_2x)\,dx = \frac-\frac+C \qquad\mbox, b_1, \neq, b_2, \mbox$
* $\int\sin^n \,dx = -\frac + \frac\int\sin^ ax\,dx \qquad\mboxn>0\mbox$
* $\int\frac = -\frac\ln+C$
* $\int\frac = \frac+\frac\int\frac \qquad\mboxn>1\mbox$
* $\begin \int x^n\sin ax\,dx &= -\frac\cos ax+\frac\int x^\cos ax\,dx \\ &= \sum_^ (-1)^ \frac\frac \cos ax +\sum_^(-1)^k \frac\frac \sin ax \\ &= - \sum_^n \frac\frac\cos\left(ax+k\frac\right) \qquad\mboxn>0\mbox \end$
* $\int\frac\,dx = \sum_^\infty (-1)^n\frac +C$
* $\int\frac\,dx = -\frac + \frac\int\frac\,dx$
* $\int\mathrm\left\}\cos\left(\right)\left(\right)\mathrm\sqrt\sqrt\sin\left(\right)\left(\right)\;\;^\mathrm\;\;}\\ & \end}\right.\;\;\;\diagup\!\!\!\!\;$
* $\int\frac = \frac\tan\left(\frac\mp\frac\right)+C$
* $\int\frac = \frac\tan\left(\frac - \frac\right)+\frac\ln\left, \cos\left(\frac-\frac\right)\+C$
* $\int\frac = \frac\cot\left(\frac - \frac\right)+\frac\ln\left, \sin\left(\frac-\frac\right)\+C$
* $\int\frac = \pm x+\frac\tan\left(\frac\mp\frac\right)+C$

Integrands involving only

cosine

* $\int\cos ax\,dx = \frac\sin ax+C$
* $\int\cos^2 \,dx = \frac + \frac \sin 2ax +C = \frac + \frac \sin ax\cos ax +C$
* $\int\cos^n ax\,dx = \frac + \frac\int\cos^ ax\,dx \qquad\mboxn>0\mbox$
* $\int x\cos ax\,dx = \frac + \frac+C$
* $\int x^2\cos^2 \,dx = \frac + \left( \frac - \frac \right) \sin 2ax + \frac \cos 2ax +C$
* $\begin \int x^n\cos ax\,dx &= \frac - \frac\int x^\sin ax\,dx \\ &= \sum_^ (-1)^ \frac\frac \cos ax +\sum_^(-1)^ \frac\frac \sin ax \\ &=\sum_^n (-1)^ \frac\frac\cos\left(ax -\frac\frac\right) \\ &=\sum_^n \frac\frac\sin\left(ax+k\frac\right) \qquad\mboxn>0\mbox \end$
* $\int\frac\,dx = \ln, ax, +\sum_^\infty (-1)^k\frac+C$
* $\int\frac\,dx = -\frac-\frac\int\frac\,dx \qquad\mboxn\neq 1\mbox$
* $\int\frac = \frac\ln\left, \tan\left(\frac+\frac\right)\+C$
* $\int\frac = \frac + \frac\int\frac \qquad\mboxn>1\mbox$
* $\int\frac = \frac\tan\frac+C$
* $\int\frac = -\frac\cot\frac+C$
* $\int\frac = \frac\tan\frac + \frac\ln\left, \cos\frac\+C$
* $\int\frac = -\frac\cot\frac+\frac\ln\left, \sin\frac\+C$
* $\int\frac = x - \frac\tan\frac+C$
* $\int\frac = -x-\frac\cot\frac+C$
* $\int(\cos a_1x)(\cos a_2x)\,dx = \frac+\frac+C \qquad\mbox, a_1, \neq, a_2, \mbox$

Integrands involving only

tangent

* $\int\tan ax\,dx = -\frac\ln, \cos ax, +C = \frac\ln, \sec ax, +C$
* $\int \tan^2 \, dx = \tan - x +C$
* $\int\tan^n ax\,dx = \frac\tan^ ax-\int\tan^ ax\,dx \qquad\mboxn\neq 1\mbox$
* $\int\frac = \frac(px + \frac\ln, q\sin ax + p\cos ax, )+C \qquad\mboxp^2 + q^2\neq 0\mbox$
* $\int\frac = \pm \frac + \frac\ln, \sin ax \pm \cos ax, +C$
* $\int\frac = \frac \mp \frac\ln, \sin ax \pm \cos ax, +C$

Integrands involving only secant

* $\int \sec \, dx = \frac\ln+C= \frac\ln+C = \frac\operatorname+C$
* $\int \sec^2 \, dx = \tan+C$
* $\int \sec^3 \, dx = \frac\sec x \tan x + \frac\ln, \sec x + \tan x, + C.$
* $\int \sec^n \, dx = \frac \,+\, \frac\int \sec^ \, dx \qquad \mboxn \ne 1\mbox$
* $\int \frac = x - \tan+C$
* $\int \frac = - x - \cot+C$

Integrands involving only

cosecant

* $\int \csc \, dx= -\frac\ln+C= \frac\ln+C = \frac\ln+C$
* $\int \csc^2 \, dx = -\cot+C$
* $\int \csc^3 \, dx = -\frac\csc x \cot x - \frac\ln, \csc x + \cot x, + C = -\frac\csc x \cot x + \frac\ln, \csc x - \cot x, + C$
* $\int \csc^n \, dx = -\frac \,+\, \frac\int \csc^ \, dx \qquad \mboxn \ne 1\mbox$
* $\int \frac = x - \frac+C$
* $\int \frac = - x + \frac+C$

Integrands involving only

cotangent

* $\int\cot ax\,dx = \frac\ln, \sin ax, +C$
* $\int \cot^2 \, dx = -\cot - x +C$
* $\int\cot^n ax\,dx = -\frac\cot^ ax - \int\cot^ ax\,dx \qquad\mboxn\neq 1\mbox$
* $\int\frac = \int\frac = \frac - \frac\ln, \sin ax + \cos ax, +C$
* $\int\frac = \int\frac = \frac + \frac\ln, \sin ax - \cos ax, +C$

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.

* $\int\frac = \frac\ln\left, \tan\left(\frac\pm\frac\right)\+C$
* $\int\frac = \frac\tan\left(ax\mp\frac\right)+C$
* $\int\frac = \frac\left(\frac + (n - 2)\int\frac \right)$
* $\int\frac = \frac + \frac\ln\left, \sin ax + \cos ax\+C$
* $\int\frac = \frac - \frac\ln\left, \sin ax - \cos ax\+C$
* $\int\frac = \frac - \frac\ln\left, \sin ax + \cos ax\+C$
* $\int\frac = -\frac - \frac\ln\left, \sin ax - \cos ax\+C$
* $\int\frac = -\frac\tan^2\frac+\frac\ln\left, \tan\frac\+C$
* $\int\frac = -\frac\cot^2\frac-\frac\ln\left, \tan\frac\+C$
* $\int\frac = \frac\cot^2\left(\frac+\frac\right)+\frac\ln\left, \tan\left(\frac+\frac\right)\+C$
* $\int\frac = \frac\tan^2\left(\frac+\frac\right)-\frac\ln\left, \tan\left(\frac+\frac\right)\+C$
* $\int(\sin ax)(\cos ax)\,dx = \frac\sin^2 ax +C$
* $\int(\sin a_1x)(\cos a_2x)\,dx = -\frac -\frac +C\qquad\mbox, a_1, \neq, a_2, \mbox$
* $\int(\sin^n ax)(\cos ax)\,dx = \frac\sin^ ax +C\qquad\mboxn\neq -1\mbox$
* $\int(\sin ax)(\cos^n ax)\,dx = -\frac\cos^ ax +C\qquad\mboxn\neq -1\mbox$
* $\begin \int(\sin^n ax)(\cos^m ax)\,dx &= -\frac+\frac\int(\sin^ ax)(\cos^m ax)\,dx \qquad\mboxm,n>0\mbox \\ &= \frac + \frac\int(\sin^n ax)(\cos^ ax)\,dx \qquad\mboxm,n>0\mbox \end$
* $\int\frac = \frac\ln\left, \tan ax\+C$
* $\int\frac = \frac+\int\frac \qquad\mboxn\neq 1\mbox$
* $\int\frac = -\frac+\int\frac \qquad\mboxn\neq 1\mbox$
* $\int\frac = \frac +C\qquad\mboxn\neq 1\mbox$
* $\int\frac = -\frac\sin ax+\frac\ln\left, \tan\left(\frac+\frac\right)\+C$
* $\int\frac = \frac-\frac\int\frac \qquad\mboxn\neq 1\mbox$
* $\begin \int \frac \, dx &= \sqrt\operatorname\left(\frac\right) - x \qquad\mbox] - \frac ; + \frac [\mbox \\ &= \sqrt\operatorname\left(\frac\right)-\operatorname\left(\tan x\right) \qquad\mbox\mbox \end$
* $\int\frac = -\frac + \int\frac \qquad\mboxn\neq 1\mbox$
* $\int\frac = \begin \frac-\frac\int\frac &\mboxm\neq 1\mbox \\ \frac-\frac\int\frac &\mboxm\neq 1\mbox \\ -\frac+\frac\int\frac &\mboxm\neq n\mbox \end$
* $\int\frac = -\frac +C\qquad\mboxn\neq 1\mbox$
* $\int\frac = \frac\left(\cos ax+\ln\left, \tan\frac\\right) +C$
* $\int\frac = -\frac\left(\frac+\int\frac\right) \qquad\mboxn\neq 1\mbox$
* $\int\frac = \begin -\frac - \frac\int\frac &\mboxn\neq 1\mbox \\ -\frac - \frac\int\frac &\mboxm\neq 1\mbox \\ \frac + \frac\int\frac &\mboxm\neq n\mbox \end$

Integrands involving both

sine and tangent

* $\int (\sin ax)(\tan ax)\,dx = \frac(\ln, \sec ax + \tan ax, - \sin ax)+C$
* $\int\frac = \frac\tan^ (ax) +C\qquad\mboxn\neq 1\mbox$

Integrand involving both

cosine and tangent

* $\int\frac = \frac\tan^ ax +C\qquad\mboxn\neq -1\mbox$

Integrand involving both

sine and cotangent

* $\int\frac = -\frac\cot^ ax +C\qquad\mboxn\neq -1\mbox$

Integrand involving both

cosine and cotangent

* $\int\frac = \frac\tan^ ax +C\qquad\mboxn\neq 1\mbox$

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

* $\int_^} \sin^n x \, dx = \int_^} \cos^n x \, dx = \frac B\left( \frac, \frac\right) = \begin \frac \cdot \frac \cdots \frac \cdot \frac \cdot \frac, & \text n\text \\ \frac \cdot \frac \cdots \frac \cdot \frac, & \text n\text \\ 1, & \text n=1 \end$

Using the modified Struve functions $L_(x)$ and modified Bessel functions $I_(x)$ one can write

* $\int_^} \exp(k \cdot \sin(x)) \, dx = \frac \left(I_0(k) + L_0(k) \right)$

Integrals with symmetric limits

* $\int_^\sin\,dx = 0$
* $\int_^\cos \,dx = 2\int_^\cos \,dx = 2\int_^\cos \,dx = 2\sin$
* $\int_^\tan \,dx = 0$
* $\int_^ x^2\cos^2 \,dx = \frac \qquad\mboxn=1,3,5...\mbox$
* $\int_^ x^2\sin^2 \,dx = \frac = \frac (1-6\frac) \qquad\mboxn=1,2,3,...\mbox$

Integral over a full circle

* $\int_^\sin^\cos^\,dx = 0 \! \qquad n,m \in \mathbb$
* $\int_^\sin^\cos^\,dx = 0 \! \qquad n,m \in \mathbb$

See also

*Trigonometric integral

References

{{DEFAULTSORT:Integrals of Trigonometric Functions
Trigonometric functions
Trigonometry