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Integration

Standard mathematical integrals

Integration

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

Integration Of The Squares Of The Circular Functions

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