Differentialrechnung

From SkyWiki

(Difference between revisions)
Jump to: navigation, search
m (Integrationsregeln)
Line 98: 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 \in[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 239: Line 136:
 +
[[category:remember]]
[[category:Mathematik]]
[[category:Mathematik]]

Revision as of 15:00, 17 September 2008

Contents

Ableitungen/Integrale

Funktion MathImage Ableitung MathImage Integral MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage
MathImage MathImage
MathImage MathImage
MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage MathImage
MathImage MathImage
MathImage MathImage
MathImage MathImage
MathImage MathImage

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

Personal tools