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$$ \int x^n \,\mathrm{d}x=\dfrac{x^{n+1}}{n+1}+k\qquad \text{if $n\neq -1$} $$$$ \int \dfrac{1}{x}\,\mathrm{d}x=\ln\vert x\vert+k $$
$$ \int e^x \,\mathrm{d}x= e^x+k $$$$ \int a^x \,\mathrm{d}x= \dfrac{a^x}{\ln a}+k $$
$$ \int \sin x \,\mathrm{d}x= -\cos x+k $$$$ \int \cos x \,\mathrm{d}x= \sin x+k $$
$$ \int \dfrac{1}{1+x^2} \,\mathrm{d}x = \tan^{-1} x+k $$$$ \int \left(1+\tan^2 x\right) \,\mathrm{d}x= \int \dfrac{1}{\cos^2 x} \,\mathrm{d}x = \tan x+k $$
$$ \dfrac{1}{\sqrt{1-x^2}} \,\mathrm{d}x = \sin^{-1} x+k $$$$ \dfrac{1}{\sqrt{1+x^2}} \,\mathrm{d}x = \sinh^{-1} x+k $$