|
m |
| (8 intermediate revisions not shown.) |
| Line 1: |
Line 1: |
| | + | == Ableitungen/Integrale == |
| | + | |
| | {|border=1 | | {|border=1 |
| | |'''Funktion <math>f(x)</math>''' | | |'''Funktion <math>f(x)</math>''' |
| Line 96: |
Line 98: |
| | |} | | |} |
| | | | |
| - | == Integrationsregeln ==
| + | siehe [[Integration]] |
| - | <math>\int[f(x)+g(x)]dx = \int f(x)dx + \int g(x)dx</math>
| + | |
| - | | + | |
| - | <math>\int{a f(x)}dx = a \int f(x)dx</math>
| + | |
| - | | + | |
| - | <math>\int[f(x)]^n f'(x)dx = \frac 1{n+1}[f(x)]^{n+1} + C,\ n\ne-1</math>
| + | |
| - | | + | |
| - | <math>\int f(x)f'(x) dx = \frac 1 2 f^2(x) + C</math>
| + | |
| - | | + | |
| - | <math>\int_a^b f(x) dx = -\int_b^a f(x) dx</math>
| + | |
| - | | + | |
| - | <math>\int_a^a f(x) dx = 0</math>
| + | |
| - | | + | |
| - | <math>\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx,\ c \epsilon [a, b]</math>
| + | |
| - | | + | |
| - | <math>\int_a^x f(t) dt = F(x)|_a^x = F(x) - F(a)\qquad</math> unbestimmtes Integral
| + | |
| - | | + | |
| - | <math>\int\frac{f'(x)}{f(x)} = \ln\left|f(x)\right|+C</math>
| + | |
| - | | + | |
| - | <math>\int\frac{f'(x)}{1+\left(f(x)\right)^2} dx = \arctan\,f(x)+C</math>
| + | |
| - | | + | |
| - | Integration durch Substitution: | + | |
| - | | + | |
| - | <math>\int f(x) dx = \int f[\varphi(t)] \dot\varphi(t) dt\quad x=\varphi(t),\ dx=\dot\varphi(t)dt \quad</math> oder
| + | |
| - | | + | |
| - | <math>\int_a^b f(x)\,dx = \int_{g^{-1}(a)}^{g^{-1}(b)} f(g(t))g'(t)\,dt,\ x = g(t)</math>
| + | |
| - | | + | |
| - | partielle Integration:
| + | |
| - | | + | |
| - | <math>\int uv'\,dx = uv - \int vu'\,dx,\ u=u(x),\ v=v(x) \quad</math> oder <math>\int u\,dv = uv-\int v\,du \quad</math> oder
| + | |
| - | | + | |
| - | <math>\int_a^b u(x) v'(x)\,dx = [u(x) v(x)]_a^b - \int_a^b v(x) u'(x)\,dx</math>
| + | |
| - | | + | |
| - | == weitere Integrale ==
| + | |
| - | {|border=1
| + | |
| - | |'''Funktion <math>f(x)</math>'''
| + | |
| - | |'''Integral <math>F(x)=\int f(x) d x</math>'''
| + | |
| - | |-
| + | |
| - | |<math>\sin^2 x</math>
| + | |
| - | |<math>\frac 1 2 (x-\sin x \cos x)</math>
| + | |
| - | |-
| + | |
| - | |<math>\cos^2 x</math>
| + | |
| - | |<math>\frac 1 2 (x+\sin x \cos x)</math>
| + | |
| - | |-
| + | |
| - | |<math>\frac 1{\cos^2 x}</math>
| + | |
| - | |<math>\tan x+C</math>
| + | |
| - | |-
| + | |
| - | |<math>\frac 1{\sin^2 x}</math>
| + | |
| - | |<math>-\cot x+C</math>
| + | |
| - | |-
| + | |
| - | |<math>\frac 1{\sqrt{1-x^2}}</math>
| + | |
| - | |<math>\arcsin x+C</math>
| + | |
| - | |-
| + | |
| - | |<math>\frac 1{\sqrt{1+x^2}}</math>
| + | |
| - | |<math>\arctan x+C</math>
| + | |
| - | |-
| + | |
| - | |<math>\frac 1{\cosh^2 x}</math>
| + | |
| - | |<math>\tanh x+C</math>
| + | |
| - | |-
| + | |
| - | |<math>\frac 1{\sinh^2 x}</math>
| + | |
| - | |<math>-\coth x+C</math>
| + | |
| - | |-
| + | |
| - | |<math>\frac 1{\sqrt{x^2+1}}</math>
| + | |
| - | |<math>\operatorname{arsinh} x+C = \ln\, \left|x+\sqrt{x^2+1}\right|+C</math>
| + | |
| - | |-
| + | |
| - | |<math>\frac 1{\sqrt{x^2-1}}</math>
| + | |
| - | |<math>\operatorname{arcosh} x+C = \ln\, \left|x+\sqrt{x^2-1}\right|+C</math>
| + | |
| - | |-
| + | |
| - | |<math>\frac 1{\sqrt{1-x^2}}</math>
| + | |
| - | |<math>\begin{cases}\operatorname{artanh}x+C=\frac 1 2 \ln\frac{1+x}{1-x}+C,& |x|<1\\\operatorname{arcoth}x+C=\frac 1 2 \ln\frac{x+1}{x-1}+C,& |x|>1\end{cases}</math>
| + | |
| - | |-
| + | |
| - | |<math>\frac 1{a^2+x^2}</math>
| + | |
| - | |<math>\frac 1 a \cdot \arctan\frac x a+C</math>
| + | |
| - | |-
| + | |
| - | |<math>\frac 1{a^2-x^2}</math>
| + | |
| - | |<math>\begin{cases}\frac 1 a \operatorname{artanh}\frac x a + C,& |x|<a\\\frac 1{2a}\ln\frac {a-x}{a+x} + C,& |x|<a\end{cases}</math>
| + | |
| - | |-
| + | |
| - | |<math>\frac 1{x^2-a^2}</math>
| + | |
| - | |<math>\begin{cases}-\frac 1 a \operatorname{arcoth}\frac x a + C,& |x|>a\\\frac 1{2a}\ln\frac {x-a}{x+a} + C\end{cases}</math>
| + | |
| - | |-
| + | |
| - | |<math>\frac 1{sqrt{a^2-x^2}}</math>
| + | |
| - | |<math>arcsin{\frac x a} + C,\ |x|<a</math>
| + | |
| - | |-
| + | |
| - | |<math>\frac 1{sqrt{a^2+x^2}}</math>
| + | |
| - | |<math>\begin{cases}\operatorname{arsinh}\frac x a + C\\\ln \left(x+\sqrt{a^2+x^2} \right)+C\end{cases}</math>
| + | |
| - | |-
| + | |
| - | |<math>\frac 1{sqrt{x^2-a^2}}</math>
| + | |
| - | |<math>\begin{cases}\operatorname{arcosh}\frac x a + C,& |x|>a\\\ln \left(x+\sqrt{x^2-a^2} \right)+C\end{cases}</math>
| + | |
| - | |-
| + | |
| - | |<math>\frac 1{sqrt{x^2 \pm a^2}}</math>
| + | |
| - | |<math>ln \left|x+\sqrt{x^2 \pm a^2} \right|+C</math>
| + | |
| - | |-
| + | |
| - | |<math>\sqrt{a^2-x^2}</math>
| + | |
| - | |<math>\frac x 2 \sqrt{a^2-x^2} + \frac{a^2}2 \arcsin\frac x a+C \quad</math>(Kreisintegral)
| + | |
| - | |-
| + | |
| - | |<math>\sqrt{a^2+x^2}</math>
| + | |
| - | |<math>\frac x 2 \sqrt{a^2+x^2} + \frac{a^2}2 \ln(x+\sqrt{a^2+x^2})+C</math>
| + | |
| - | |-
| + | |
| - | |<math>(a x + b)^n</math>
| + | |
| - | |<math>\frac{(a x + b)^{n+1}}{a(n+1)},\ n\ne-1</math>
| + | |
| - | |-
| + | |
| - | |<math>\frac 1{ax+b}</math>
| + | |
| - | |<math>\frac 1 a \ln|ax+b|</math>
| + | |
| - | |}
| + | |
| | | | |
| | == Integralfunktionen == | | == Integralfunktionen == |
| Line 221: |
Line 120: |
| | | | |
| | | | |
| - | == Fourier-Transformation == | + | == [http://de.wikipedia.org/wiki/Fourier-Transformation Fourier-Transformation] == |
| | + | |
| | + | Bildfunktion <math>F(j\omega) = \mathcal F\{f(t)\} = \int_{-\infty}^\infty f(t)\, e^{-j\omega t} dt</math> |
| | + | |
| | + | |
| | + | == [http://de.wikipedia.org/wiki/Laplace-Transformation Laplace-Transformation] == |
| | + | |
| | + | Originalfunktion, Oberfunktion: |
| | + | |
| | + | <math>f(t) = \mathcal L^{-1}\{f(t)\} = \frac 1 {2j\pi} \int_{\beta-j\omega}^{\beta+j\omega} F(s)\, e^{s t} ds</math> |
| | + | |
| | + | Laplace-Transformierte, Unterfunktion, Bildfunktion: |
| | | | |
| - | Bildfunktion <math>F(j\omega) = \mathcal {F}\{f(t)\} = \int_{-\infty}^{\infty} f(t) e^{-j\omega t}</math>
| + | <math>F(s) = \mathcal L\{f(t)\} = \int_0^\infty f(t)\, e^{-s t} dt</math> |
| | | | |
| | | | |
| | + | [[category:remember]] |
| | [[category:Mathematik]] | | [[category:Mathematik]] |
