Integrals of Trigonometric Functions

List of integrals involving trigonometric functions

\( \int \sin x ~  dx = -\cos x \)

 

\( \int \cos x ~ dx = \sin x \)

 

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

 

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

 

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

 

\( \int \cos^3x ~ dx = \sin x – \frac{1}{3}\sin^3x \)

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

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

 

\( \int \tan x~dx = -\ln|\cos x| \)

 

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

 

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

 

\( \int \tan^2x~dx = \tan x-x \)

 

\( \int \cot x~dx =\ln|\sin x| \)

 

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

 

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

 

\( \int \cot^2x~dx = -\cot x – x \)

 

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

 

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

 

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

 

\( \int \frac{dx}{\sin^2x \cdot \cos^2x}=\tan x – \cot x \)

 

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

 

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

 

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

 

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

 

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

 

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

 

\( \int \arccos x~dx = x \cdot \arccos x – \sqrt{1-x^2} \)

 

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

 

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

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