# Integral table

## Inverse derivatives

$\int \mathrm{d}x = x + C$ $\int 0\mathrm{d}x = C$ $\int x^m\mathrm{d}x = \frac{x^{m+1}}{m+1} + C, m \neq -1$ $\int \frac{\mathrm{d}x}{x} = \ln|x| + C$ $\int \cos(x) \mathrm{d}x = \sin(x) + C$ $\int \sin(x) \mathrm{d}x = -\cos(x) + C$ $\int \frac{\mathrm{d}x}{1+x^2} = \arctg(x) + C = -\arcctg(x) + C$ $\int \frac{\mathrm{d}x}{\sqrt{1-x^2}} = \arcsin(x) + C = -\arccos(x) + C$ $\int a^x \mathrm{d}x = \frac{a^x}{\ln a} + C$ $\int e^x \mathrm{d}x = e^x + C$ $\int \sec(x)^2 \mathrm{d}x = \int \frac{dx}{\cos(x)^2} = \tg(x) + C$ $\int \cosec(x)^2 \mathrm{d}x = \int \frac{dx}{\sin(x)^2} = -\ctg(x) + C$ $\int \sh(x)\mathrm{d}x = \ch(x) + C$ $\int \ch(x)\mathrm{d}x = \sh(x) + C$ $\int \frac{\mathrm{d}x}{\ch(x)^2} = \th(x) + C$ $\int \frac{\mathrm{d}x}{\sh(x)^2} = -\cth(x) + C$

## Standart integrals

$\int \frac{\mathrm{d}x}{x^2 + a^2} = \frac{1}{a} \arctg\Big(\frac{x}{a}\Big) + C$ $\int \frac{\mathrm{d}x}{x^2 - a^2} = \frac{1}{2a}\ln\Big|\frac{x-a}{x+a}\Big| + C$ $\int \frac{\mathrm{d}x}{\sqrt{x^2 \pm a^2}} = \ln|x + \sqrt{x^2 \pm a^2}| + C$ $\int \frac{\mathrm{d}x}{\sqrt{a^2 - x^2}} = \arcsin \Big(\frac{x}{a}\Big) + C$