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MathsCalculus

Integration

Standard mathematical integrals

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

Last Modified: 10 Feb 14 @ 14:02     Page Rendered: 2022-03-16 07:47:52