Differentialrechnung

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Ableitungen/Integrale

Funktion $MathImage$	Ableitung $MathImage$	Integral $MathImage$
$MathImage$	$MathImage$	$MathImage$
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$MathImage$	$MathImage$	$MathImage$
$MathImage$	$MathImage$	$MathImage$
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$MathImage$	$MathImage$	$MathImage$
$MathImage$	$MathImage$	$MathImage$
$MathImage$	$MathImage$	$MathImage$
$MathImage$	$MathImage$	$MathImage$
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$MathImage$	$MathImage$	$MathImage$
$MathImage$	$MathImage$	$MathImage$
$MathImage$	$MathImage$	$MathImage$
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siehe Integration

Integralfunktionen

Integralsinus	$MathImage$	Eulersche Konstante $MathImage$
Integralcosinus	$MathImage$
Exponentialintegral	$MathImage$
Integrallogarithmus	$MathImage$
Gaußsches Fehlerintegral	$MathImage$

Fourier-Transformation

Bildfunktion $MathImage$

Laplace-Transformation

Originalfunktion, Oberfunktion:

$MathImage$

Laplace-Transformierte, Unterfunktion, Bildfunktion:

$MathImage$

@@ Line 1: / Line 1: @@
+== Ableitungen/Integrale ==
 {|border=1
 |'''Funktion <math>f(x)</math>'''
@@ Line 19: / Line 21: @@
 |<math>-\frac 1 {x^2}</math>
 |<math>\ln\, \left|x\right|+C</math>
+|-
+|<math>\frac 1{ax+b}</math>
+|
+|<math>\frac 1 a \ln|ax+b|+C</math>
 |-
 |<math>e^x</math>
@@ Line 30: / Line 36: @@
 |<math>\ln x</math>
 |<math>\frac 1 x</math>
+|<math>-x + x\,\ln\,x+C</math>
 |-
 |<math>\log_a x</math>
@@ Line 44: / Line 51: @@
 |<math>\tan x</math>
 |<math>\frac 1 {\cos^2 x} = 1+\tan^2 x</math>
+|<math>-\ln|\cos\,x|+C</math>
 |-
 |<math>\cot x</math>
 |<math>-\frac 1 {\sin^2 x} = -(1+\cot^2 x)</math>
+|<math>\ln|\sin\,x|+C</math>
 |-
 |<math>\arcsin x</math>
@@ Line 89: / 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 f(x) dx = \int f[\varphi(t)] \dot\varphi(t) dt\qquad</math> Integration durch Substitution, <math>x=\varphi(t),\ dx=\dot\varphi(t)dt</math>
-<math>\int uv'\,dx = uv - \int vu'\,dx,\ u=u(x),\ v=v(x)\qquad</math> oder <math>\int u\,dv = uv-\int v\,du\qquad</math>partielle Integration
-== weitere Integrale ==
-{|border=1
-|'''Funktion <math>f(x)</math>'''
-|'''Integral <math>F(x)=\int f(x) d 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 \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,& |x|<a\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>
-|}
 == Integralfunktionen ==
 {|
 |Integralsinus
-|<math>Si(x) = \int_0^x\frac{\sin t}t dt = </math>
+|<math>Si(x) = \int_0^x\frac{\sin\,t}t dt = C+\ln|x|-\frac{x^2}{2\cdot 2!}+\frac{x^4}{4\cdot 4!}-\frac{x^6}{6\cdot 6!} + - \cdots,\ |x|<\infty</math>
+|Eulersche Konstante <math>C =0,5772...</math>
 |-
 |Integralcosinus
-|<math>Ci(x) = \int_x^\infty\frac{\cos t}t dt = </math>
+|<math>Ci(x) = \int_x^\infty\frac{\cos\,t}t dt = C+\ln|x|+\frac{x}{1\cdot 1!}+\frac{x^2}{2\cdot 2!}+\frac{x^3}{3\cdot 3!} + \cdots,\ |x|<\infty</math>
 |-
-|Exponetialintegral
+|Exponentialintegral
-|<math>Ei(x) = \int_{-\infty}^x\frac{e^t}t dt = </math>
+|<math>Ei(x) = \int_{-\infty}^x\frac{e^t}t dt = C+\ln|x|+\frac x{1\cdot 1!}+\frac{x^2}{2\cdot 2!}+\frac{x^3}{3\cdot 3!} + \cdots,\ x<0</math>
 |-
 |Integrallogarithmus
-|<math>Li(x) = \int_0^x\frac{dt}{\ln t}t = </math>
+|<math>Li(x) = \int_0^x\frac{dt}{\ln\,t}t = \int_{-\infty}^x\frac{e^t}t dt = C+\ln\left|\ln x\right|+\frac{\ln x}{1\cdot 1!}+\frac{(\ln\,x)^2}{2\cdot 2!}-\frac{(\ln\,x)^3}{3\cdot 3!} + \cdots,\ 0<x<\infty</math>
 |-
 |Gaußsches Fehlerintegral
-|<math>F(x) = \frac 1{\sqrt{2\pi}}\int_-\infty^xe^{-\frac{t^2} 2} dt = </math>
+|<math>F(x) = \frac 1{\sqrt{2\pi}} \int_{-\infty}^xe^{-\frac{t^2} 2} dt = \frac 1 2 + \frac 1{\sqrt{2\pi}} \left(\frac x 1-\frac{x^3}{2\cdot3\cdot1!}+\frac{x^5}{2^2\cdot5\cdot2!}-\frac{x^7}{2^3\cdot7\cdot3!}+-\cdots\right),\ |x|<\infty</math>
 |}
+== [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:
+<math>F(s) = \mathcal L\{f(t)\} = \int_0^\infty f(t)\, e^{-s t} dt</math>
+[[category:remember]]
 [[category:Mathematik]]

Differentialrechnung

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Ableitungen/Integrale

Integralfunktionen

Fourier-Transformation

Laplace-Transformation

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