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Maths › Calculus ›

Integration

Standard mathematical integrals

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Contents

Integration
Integration Of The Squares Of The Circular Functions
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Integration

$x^n\:dx$	$= \frac{x^{n\,+\,1}}{n+1}$ for all values of n except n = - 1
$\frac{1}{x}\:dx$	$= \ln\:x$
$e^x\:dx$	$= e^x$
$\int_{}^{}\sin\:x\:dx$	$= -\cos\:x$
$\int_{}^{}\cos\:x\:dx$	$= \;\sin\:x$
$\int \tan\:x\:dx$	$= -\ln \cos x$
$\int \sec^2\:x\:dx$	$= tan\:x$
$\int \frac{1}{a^2 + x^2}\:dx$	$= \frac{1}{a} \tan^{-1}\frac{x}{a}$
$\int \frac{1}{a^2 - x^2}\:dx$	$= \frac{1}{2a} \ln \frac{a + x}{a - x} \frac{1}{a} \tanh^{-1}\frac{x}{a}$
$\int \frac{1}{x^2 - a^2}\:dx$	$= \frac{1}{2\,a} \ln \left(\frac{x-a}{x+a} \right) = -\frac{1}{a} \coth^{-1}\frac{x}{a}$
$\int \frac{1}{\sqrt[]{(a^2}-x^2)}\:dx$	$= \sin^{-1} \frac{x}{a}$
$\int \frac{1}{\sqrt[]{(a^2\:+\:x^2})}\:dx$	$= \ln\left(x + \sqrt{(x^2\:+\:a^2)} \right) = \sinh^{-1} \frac{x}{a}$
$\int \frac{1}{\sqrt]{(x^2\:-\:a^2)}}\:dx$	$= Ln\left(x\:+\:\sqrt{(x^2\:-\:a^2}) \right)$

Integration Of The Squares Of The Circular Functions

$\int \sin^2 x\:dx$	$= \frac{1}{2} x - \frac{1}{4}\:\sin 2x$
$\int \cos^2 x\:dx$	$= \frac{1}{2}x + \frac{1}{4} \sin\:2\,x$
$\int \tan^2\:x\:dx$	$= (\tan x) - x$
$\int \cot^2 x\:dx$	$= -\:(\cot x)-x$
$\int \cosec\:x\:dx$	$= -\cot x$