Differentialrechnung

From SkyWiki

Revision as of 10:14, 17 September 2008 (view source) Martin (Talk | contribs) (→Integrationsregeln) ← Previous diff		Current revision (16:52, 20 May 2011) (view source) Martin-Admin (Talk | contribs) m (Reverted edits by SallyJo (Talk) to last version by Martin)
(15 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 19:		Line 21:
	|<math>-\frac 1 {x^2}</math>		|<math>-\frac 1 {x^2}</math>
	|<math>\ln\, \left|x\right|+C</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>		|<math>e^x</math>
Line 92:		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>\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>\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></math>	+
-	|<math></math>	+
-	|-	+
-	|<math></math>	+
-	|<math></math>	+
-	|-	+
-	|<math></math>	+
-	|<math></math>	+
-	|-	+
-	|<math></math>	+
-	|<math></math>	+
-	|}	+
	== Integralfunktionen ==		== Integralfunktionen ==
Line 214:		Line 120:
		+	== [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]]		[[category:Mathematik]]

---

Current revision

Contents 1 Ableitungen/Integrale 2 Integralfunktionen 3 Fourier-Transformation 4 Laplace-Transformation

Ableitungen/Integrale

Funktion	Ableitung	Integral

siehe Integration

Integralfunktionen

Integralsinus		Eulersche Konstante
Integralcosinus
Exponentialintegral
Integrallogarithmus
Gaußsches Fehlerintegral

Fourier-Transformation

Bildfunktion

Laplace-Transformation

Originalfunktion, Oberfunktion:

Laplace-Transformierte, Unterfunktion, Bildfunktion: