| Konstante Funktionen |
\(\int 0 ~ \mathrm{d}x = 0 \cdot \int \mathrm{d}x\) |
\(0 \cdot x + C = C\) |
|
| \(\int 1 ~ \mathrm{d}x = 1 \cdot \int \mathrm{d}x\) |
\(1 \cdot x + C = x + C\) |
|
| \(\int k ~ \mathrm{d}x = k \cdot \int \mathrm{d}x\) |
\(k \cdot x + C\) |
|
| Potenzfunktionen |
\(\int x^n~\mathrm{d}x\) |
\(\dfrac{x^{n+1}}{n+1}+C\) |
für \(n \neq -1\) |
| \(\int x^{-1}~\mathrm{d}x = {\displaystyle \int} \dfrac{1}{x} ~\mathrm{d}x\) |
\(\ln|x|+C\) |
für \(x \neq 0\) |
| \(\int x ~\mathrm{d}x\) |
\(\dfrac{1}{2}x^2 + C\) |
|
| \(\int 2x ~\mathrm{d}x\) |
\(x^2 + C\) |
|
| \(\int x^2 ~\mathrm{d}x\) |
\(\dfrac{1}{3}x^3 + C\) |
|
| \(\int 3x^2 ~\mathrm{d}x\) |
\(x^3 + C\) |
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| \({\displaystyle \int} -\dfrac{2}{x^3} ~\mathrm{d}x\) |
\(\dfrac{1}{x^2} + C\) |
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| \({\displaystyle \int} -\dfrac{1}{x^2} ~\mathrm{d}x\) |
\(\dfrac{1}{x} + C\) |
|
| Wurzelfunktionen |
\(\int \sqrt{x}~\mathrm{d}x\) |
\(\dfrac{2}{3} x^{\frac{3}{2}}+C\) |
|
| \(\int \sqrt[n]{x}~\mathrm{d}x\) |
\(\dfrac{n}{n+1} \left(\sqrt[n]{x}\right)^{n+1}+C\) |
für \(n \neq -1\) |
| \({\displaystyle \int} \dfrac{1}{\sqrt{x}}~\mathrm{d}x\) |
\(2 \sqrt{x}+C\) |
|
| \({\displaystyle \int} \dfrac{1}{n\left(\sqrt[n]{x^{n-1}}\right)}~\mathrm{d}x\) |
\(\sqrt[n]{x}+C\) |
|
| Exponentialfunktionen |
\(\int e^x ~\mathrm{d}x\) |
\(e^x+C\) |
|
| \(\int e^{\alpha x} ~\mathrm{d}x\) |
\(\dfrac{1}{\alpha}e^{\alpha x}+C\) |
|
| \(\int a^{x} ~\mathrm{d}x\) |
\(\dfrac{a^x}{\ln(a)}+C\) |
für \(a \in \mathbb{R}^+ / \{1\}\) |
| Logarithmusfunktionen |
\(\int \ln(x) ~\mathrm{d}x\) |
\(x \cdot \ln(x)-x+C\) |
|
| \(\int \log_a x ~\mathrm{d}x\) |
\(\dfrac{1}{\ln(a)}(x \cdot \ln(x) - x)+C\) |
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| \(\int a^{x} \cdot \ln(a) ~\mathrm{d}x\) |
\(a^x+C\) |
für \(a \in \mathbb{R}^+\) |
| Trigonometrische Funktionen |
\(\int \sin(x) ~\mathrm{d}x\) |
\(- \cos(x)+C\) |
|
| \(\int \cos(x) ~\mathrm{d}x\) |
\(\sin(x)+C\) |
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| \({\displaystyle \int} \dfrac{1}{\sin^2(x)} ~\mathrm{d}x\) |
\(- \cot(x)+C\) |
\(x \neq k\pi\) mit \(k \in \mathbb{Z}\) |
| \({\displaystyle \int} \dfrac{1}{\cos^2(x)} ~\mathrm{d}x\) |
\(\tan(x)+C\) |
\(x \neq \dfrac{\pi}{2}+k\pi\) mit \(k \in \mathbb{Z}\) |
| Hyperbelfunktionen |
\(\int \sinh(x) ~\mathrm{d}x\) |
\(\cosh(x)+C\) |
|
| \(\int \cosh(x) ~\mathrm{d}x\) |
\(\sinh(x)+C\) |
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| \({\displaystyle \int} \dfrac{1}{\sinh^2(x)} ~\mathrm{d}x\) |
\(- \coth(x)+C\) |
\(x \neq 0\) |
| \({\displaystyle \int} \dfrac{1}{\cosh^2(x)} ~\mathrm{d}x\) |
\(\tanh(x)+C\) |
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