Integral table

Inverse derivatives

dx=x+C\int \mathrm{d}x = x + C 0dx=C\int 0\mathrm{d}x = C xmdx=xm+1m+1+C,m1\int x^m\mathrm{d}x = \frac{x^{m+1}}{m+1} + C, m \neq -1 dxx=lnx+C\int \frac{\mathrm{d}x}{x} = \ln|x| + C cos(x)dx=sin(x)+C\int \cos(x) \mathrm{d}x = \sin(x) + C sin(x)dx=cos(x)+C\int \sin(x) \mathrm{d}x = -\cos(x) + C dx1+x2=arctg(x)+C=arcctg(x)+C\int \frac{\mathrm{d}x}{1+x^2} = \arctg(x) + C = -\arcctg(x) + C dx1x2=arcsin(x)+C=arccos(x)+C\int \frac{\mathrm{d}x}{\sqrt{1-x^2}} = \arcsin(x) + C = -\arccos(x) + C axdx=axlna+C\int a^x \mathrm{d}x = \frac{a^x}{\ln a} + C exdx=ex+C\int e^x \mathrm{d}x = e^x + C sec(x)2dx=dxcos(x)2=tg(x)+C\int \sec(x)^2 \mathrm{d}x = \int \frac{dx}{\cos(x)^2} = \tg(x) + C cosec(x)2dx=dxsin(x)2=ctg(x)+C\int \cosec(x)^2 \mathrm{d}x = \int \frac{dx}{\sin(x)^2} = -\ctg(x) + C sh(x)dx=ch(x)+C\int \sh(x)\mathrm{d}x = \ch(x) + C ch(x)dx=sh(x)+C\int \ch(x)\mathrm{d}x = \sh(x) + C dxch(x)2=th(x)+C\int \frac{\mathrm{d}x}{\ch(x)^2} = \th(x) + C dxsh(x)2=cth(x)+C\int \frac{\mathrm{d}x}{\sh(x)^2} = -\cth(x) + C

Standart integrals

dxx2+a2=1aarctg(xa)+C\int \frac{\mathrm{d}x}{x^2 + a^2} = \frac{1}{a} \arctg\Big(\frac{x}{a}\Big) + C dxx2a2=12alnxax+a+C\int \frac{\mathrm{d}x}{x^2 - a^2} = \frac{1}{2a}\ln\Big|\frac{x-a}{x+a}\Big| + C dxx2±a2=lnx+x2±a2+C\int \frac{\mathrm{d}x}{\sqrt{x^2 \pm a^2}} = \ln|x + \sqrt{x^2 \pm a^2}| + C dxa2x2=arcsin(xa)+C\int \frac{\mathrm{d}x}{\sqrt{a^2 - x^2}} = \arcsin \Big(\frac{x}{a}\Big) + C

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