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Integrals of Trigonometric Functions

List of integrals involving trigonometric functions

\int sin x d x = - cos x

$\int \sin x ~ dx = -\cos x$

\int cos x d x = sin x

$\int \cos x ~ dx = \sin x$

\int sin 2 x d x = x 2 - 1 4 sin (2 x)

$\int \sin^2x ~ dx= \frac{x}{2}-\frac{1}{4}\sin(2x)$

\int cos 2 x d x = x 2 + 1 4 sin (2 x)

$\int \cos^2x ~ dx = \frac{x}{2}+\frac{1}{4}\sin(2x)$

\int sin 3 x d x = 1 3 cos 3 x - cos x

$\int \sin^3x ~ dx = \frac{1}{3}\cos^3x-\cos x$

\int cos 3 x d x = sin x - 1 3 sin 3 x

$\int \cos^3x ~ dx = \sin x - \frac{1}{3}\sin^3x$

\int d x sin x = ln ∣ ∣ tan x 2 ∣ ∣

$\int \frac{dx}{\sin x} = \ln\left| \tan \frac{x}{2} \right|$

\int d x cos x = ln ∣ ∣ ∣ tan (x 2 + π 4) ∣ ∣ ∣

$\int \frac{dx}{\cos x} = \ln\left| \tan \left(\frac{x}{2} + \frac{\pi}{4}\right)\right|$

\int d x sin 2 x = - cot x

$\int \frac{dx}{\sin^2x} = -\cot x$

\int d x cos 2 x = tan x

$\int \frac{dx}{\cos^2x} = \tan x$

\int d x sin 3 x = - cos x 2 \cdot sin 2 x + 1 2 ln ∣ ∣ tan x 2 ∣ ∣

$\int \frac{dx}{\sin^3x} = -\frac{\cos x}{2\cdot \sin^2x} + \frac{1}{2}\ln\left|\tan\frac{x}{2}\right|$

\int d x cos 3 x = sin x 2 \cdot cos 2 x + 1 2 ln ∣ ∣ ∣ tan (x 2 + π 2) ∣ ∣ ∣

$\int \frac{dx}{\cos^3x} = \frac{\sin x}{2\cdot \cos^2x} + \frac{1}{2}\ln\left|\tan\left(\frac{x}{2}+\frac{\pi}{2}\right)\right|$

\int sin x \cdot cos x d x = - 1 4 cos (2 x)

$\int \sin x \cdot \cos x dx = - \frac{1}{4}\cos(2x)$

\int sin 2 x \cdot cos x d x = 1 3 sin 3 x

$\int \sin^2x \cdot \cos x dx = \frac{1}{3}\sin^3x$

\int sin x \cdot cos 2 x d x = - 1 3 cos 3 x

$\int \sin x \cdot \cos^2x dx = -\frac{1}{3}\cos^3x$

\int sin 2 x \cdot cos 2 x d x = x 8 - 1 32 sin (4 x)

$\int \sin^2x \cdot \cos^2x dx = \frac{x}{8}-\frac{1}{32}\sin(4x)$

\int tan x d x = - ln | cos x |

$\int \tan x~dx = -\ln|\cos x|$

\int sin x cos 2 x d x = 1 cos x

$\int \frac{\sin x}{\cos^2x}dx = \frac{1}{\cos x}$

\int sin 2 x cos x d x = ln ∣ ∣ ∣ tan (x 2 + π 4) ∣ ∣ ∣ - sin x

$\int \frac{\sin^2x}{\cos x} dx = \ln \left| \tan\left( \frac{x}{2}+\frac{\pi}{4}\right ) \right| - \sin x$

\int tan 2 x d x = tan x - x

$\int \tan^2x~dx = \tan x-x$

\int cot x d x = ln | sin x |

$\int \cot x~dx =\ln|\sin x|$

\int cos x sin 2 x d x = - 1 sin x

$\int \frac{\cos x}{\sin^2x}dx=-\frac{1}{\sin x}$

\int cos 2 x sin x d x = ln ∣ ∣ tan x 2 ∣ ∣ + cos x

$\int \frac{\cos^2x}{\sin x}dx = \ln\left|\tan\frac{x}{2}\right| + \cos x$

\int cot 2 x d x = - cot x - x

$\int \cot^2x~dx = -\cot x - x$

\int d x sin x \cdot cos x = ln | tan x |

$\int \frac{dx}{\sin x \cdot \cos x} = \ln|\tan\,x|$

\int d x sin 2 x \cdot cos x = - 1 sin x + ln ∣ ∣ ∣ tan (x 2 + π 4) ∣ ∣ ∣

$\int \frac{dx}{\sin^2x \cdot \cos x} = -\frac{1}{\sin x} + \ln\left|\tan\left(\frac{x}{2} + \frac{\pi}{4}\right)\right|$

\int d x sin x \cdot cos 2 x = 1 cos x + ln ∣ ∣ tan x 2 ∣ ∣

$\int \frac{dx}{\sin x \cdot \cos^2x}=\frac{1}{\cos x}+\ln\left|\tan\frac{x}{2}\right|$

\int d x sin 2 x \cdot cos 2 x = tan x - cot x

$\int \frac{dx}{\sin^2x \cdot \cos^2x}=\tan x - \cot x$

\int sin (m x) \cdot sin (n x) d x = - sin ( m + n ) x 2 ( m + n ) + sin ( m - n ) x 2 ( m - n ), m 2 \neq n 2

$\int \sin(mx)\cdot \sin(nx)~dx = -\frac{\sin(m+n)x}{2(m+n)} + \frac{\sin(m-n)x}{2(m-n)}, \quad m^2 \ne n^2$

\int sin (m x) \cdot cos (n x) d x = - cos ( m + n ) x 2 ( m + n ) - cos ( m - n ) x 2 ( m - n ), m 2 \neq n 2

$\int \sin(mx)\cdot \cos(nx)~dx = -\frac{\cos(m+n)x}{2(m+n)} - \frac{\cos(m-n)x}{2(m-n)}, \quad m^2 \ne n^2$

\int cos (m x) \cdot cos (n x) d x = sin ( m + n ) x 2 ( m + n ) + sin ( m - n ) x 2 ( m - n ), m 2 \neq n 2

$\int \cos(mx)\cdot \cos(nx)~dx = \frac{\sin(m+n)x}{2(m+n)} + \frac{\sin(m-n)x}{2(m-n)}, \quad m^2 \ne n^2$

\int sin x \cdot cos n x d x = sin n + 1 x n + 1

$\int \sin x \cdot \cos^nx~dx = \frac{\sin^{n+1}x}{n+1}$

\int sin n x \cdot cos x d x = sin n + 1 x n + 1

$\int \sin^nx \cdot \cos\,x~dx = \frac{\sin^{n+1}x}{n+1}$

\int arcsin x d x = x \cdot arcsin x + 1 - x 2 - - - - - \sqrt

$\int \arcsin x~dx = x\cdot \arcsin x + \sqrt{1-x^2}$

\int arccos x d x = x \cdot arccos x - 1 - x 2 - - - - - \sqrt

$\int \arccos x~dx = x \cdot \arccos x - \sqrt{1-x^2}$

\int arctan x d x = x \cdot arctan x - 1 2 ln (1 + x 2)

$\int \arctan x ~dx = x \cdot \arctan x - \frac{1}{2}\ln(1+x^2)$

\int a r c c o t x d x = x \cdot a r c c o t x + 1 2 ln (1 + x 2)

$\int \mathrm{arccot}\,x~dx = x \cdot \mathrm{arccot}\,x + \frac{1}{2}\ln(1+x^2)$

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