Math Indefinite integrals Integrating/Integration

# Integrating/Integration

Forming the antiderivative of a given function is called integrating or integration

## Integration rules

Similar to the differentiation rules, there are also integration rules when integrating indefinite integrals.

 constant-, power- and constant factor rule $\int k\,\mathrm{d}x=kx+C$ $\int x^n \,\mathrm{d}x=$ $\frac{1}{n+1}x^{n+1}+C$ $\int a \cdot g(x) \, \mathrm{d}x =$ $a \cdot \int g(x) \, \mathrm{d}x$ sum rule $\int f(x)+g(x) \, \mathrm{d}x =$ $\int f(x) \, \mathrm{d}x + \int g(x) \, \mathrm{d}x$ integration by parts $\int f(x) g'(x) \, \mathrm{d}x =$ $f(x) g(x) - \int f'(x) g(x) \, \mathrm{d}x$ integration by linear substitution $\int f(mx+n) \, \mathrm{d}x=$ $\frac1m F(mx+n)+C$

## Important integrals

Here are some important integrals of elementary functions to avoid a complicated derivation.

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### Hint

It is a good idea to memorize these antiderivatives, as they occur more frequently in exercises.
Function f(x)Antiderivative F(x)
power- and root functions
$f(x)=x^n$ $\int x^n \,\mathrm{d}x=\frac{1}{n+1}x^{n+1}+C$
$f(x)=x^{-1}=\frac1x$ $\int \frac1x \,\mathrm{d}x=\ln|x|+C$
$f(x)=\sqrt{x}$ $\int \sqrt{x} \,\mathrm{d}x=\frac23\sqrt{x^3}+C$
sine and cosine functions
$f(x)=\sin(x)$ $\int \sin(x) \,\mathrm{d}x=$ $-\cos(x)+C$
$f(x)=\cos(x)$ $\int \cos(x) \,\mathrm{d}x=$ $\sin(x)+C$
exponential functions
$f(x)=a^x$ $\int a^x \,\mathrm{d}x=\frac{a^x}{\ln(a)}+C$
$f(x)=e^x$ $\int e^x \,\mathrm{d}x=e^x+C$