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 | ||
| $f(x)=a^x$ | $\int a^x \,\mathrm{d}x=\frac{a^x}{\ln(a)}+C$ | |
| $f(x)=e^x$ | $\int e^x \,\mathrm{d}x=e^x+C$ | |