Tabelle von Ableitungs- und Stammfunktionen

Diese Tabelle v​on Ableitungs- u​nd Stammfunktionen (Integraltafel) g​ibt eine Übersicht über Ableitungsfunktionen u​nd Stammfunktionen, d​ie in d​er Differential- u​nd Integralrechnung benötigt werden.

Tabelle einfacher Ableitungs- und Stammfunktionen (Grundintegrale)

Diese Tabelle i​st zweispaltig aufgebaut. In d​er linken Spalte s​teht eine Funktion, i​n der rechten Spalte e​ine Stammfunktion dieser Funktion. Die Funktion i​n der linken Spalte i​st somit d​ie Ableitung d​er Funktion i​n der rechten Spalte.

Hinweise:

• Wenn ${\displaystyle F}$ eine Stammfunktion von ${\displaystyle f}$ ist und ${\displaystyle C}$ eine beliebige reelle Zahl (Konstante), dann ist auch ${\displaystyle F(x)+C}$ eine Stammfunktion von ${\displaystyle f}$. Zum Beispiel ist auch ${\displaystyle F(x)={\tfrac {1}{2}}x^{2}+5}$ eine Stammfunktion von ${\displaystyle f(x)=x}$. Ist der Definitionsbereich von ${\displaystyle f}$ ein Intervall, so erhält man auf diese Art alle Stammfunktionen. Besteht der Definitionsbereich von ${\displaystyle f}$ aus mehreren Intervallen, so kann die additive Konstante auf jedem der Intervalle getrennt gewählt werden. Die additive Konstante ${\displaystyle C}$ wird aus Gründen der Übersichtlichkeit in der Tabelle nicht aufgeführt.
• Weiterhin gilt: Falls ${\displaystyle F(x)}$ eine Stammfunktion von ${\displaystyle f(x)}$ ist, so ist aufgrund der Linearität des Integrals ${\displaystyle a\cdot F(x)}$ eine Stammfunktion von ${\displaystyle a\cdot f(x)}$.
• Ebenso gilt: Sind ${\displaystyle F(x)}$ und ${\displaystyle G(x)}$ Stammfunktionen von ${\displaystyle f(x)}$ und ${\displaystyle g(x)}$, so ist ${\displaystyle F(x)+G(x)}$ eine Stammfunktion von ${\displaystyle f(x)+g(x)}$.

Potenz- und Wurzelfunktionen

Funktion ${\displaystyle f(x)}$	Stammfunktion ${\displaystyle F(x)}$
${\displaystyle 0}$	${\displaystyle 0}$
${\displaystyle k\quad (k\in \mathbb {R} )}$	${\displaystyle kx}$
${\displaystyle x^{n}}$	${\displaystyle {\begin{cases}{\frac {1}{n+1}}x^{n+1},&{\text{wenn }}n\neq -1,\\\ln |x|,&{\text{wenn }}n=-1.\end{cases}}}$
${\displaystyle nx^{n-1}}$	${\displaystyle x^{n}}$
${\displaystyle x}$	${\displaystyle {\tfrac {1}{2}}x^{2}}$
${\displaystyle 2x}$	${\displaystyle x^{2}}$
${\displaystyle x^{2}}$	${\displaystyle {\tfrac {1}{3}}x^{3}}$
${\displaystyle 3x^{2}}$	${\displaystyle x^{3}}$
${\displaystyle {\sqrt {x}}}$	${\displaystyle {\tfrac {2}{3}}x^{\tfrac {3}{2}}}$
${\displaystyle {\sqrt[{n}]{x}}}$	${\displaystyle {\frac {n}{n+1}}({\sqrt[{n}]{x}})^{n+1}\quad (n\neq -1)}$
${\displaystyle {\frac {1}{\sqrt {x}}}}$	${\displaystyle 2{\sqrt {x}}}$
${\displaystyle {\frac {1}{n({\sqrt[{n}]{x^{n-1}}})}}}$	${\displaystyle {\sqrt[{n}]{x}}}$
${\displaystyle -{\frac {2}{x^{3}}}}$	${\displaystyle {\frac {1}{x^{2}}}}$
${\displaystyle -{\frac {1}{x^{2}}}}$	${\displaystyle {\frac {1}{x}}}$

Exponential- und Logarithmusfunktionen

Funktion ${\displaystyle f(x)}$	Stammfunktion ${\displaystyle F(x)}$
${\displaystyle \mathrm {e} ^{x}}$	${\displaystyle \mathrm {e} ^{x}}$
${\displaystyle \mathrm {e} ^{kx}}$	${\displaystyle {\frac {1}{k}}\mathrm {e} ^{kx}}$
${\displaystyle a^{x}\ln a\quad (a>0)}$	${\displaystyle a^{x}}$
${\displaystyle a^{x}}$	${\displaystyle {\frac {a^{x}}{\ln a}}}$
${\displaystyle x^{x}(1+\ln(x))}$	${\displaystyle x^{x}\quad (x>0)}$
${\displaystyle \mathrm {e} ^{x\ln \left|x\right|}(\ln \left|x\right|+1)}$	${\displaystyle \left|x\right|^{x}=\mathrm {e} ^{x\ln \left|x\right|}\quad (x\neq 0)}$
${\displaystyle {\frac {1}{x}}}$	${\displaystyle \ln \left|x\right|}$[A 1]
${\displaystyle \ln x}$	${\displaystyle x\ln(x)-x}$
${\displaystyle x^{n}\ln x}$	${\displaystyle {\frac {x^{n+1}}{n+1}}\left(\ln x-{\frac {1}{n+1}}\right)\quad (n\geq 0)}$
${\displaystyle u'(x)\ln u(x)}$	${\displaystyle u(x)\ln u(x)-u(x)}$
${\displaystyle {\frac {1}{x}}\ln ^{n}x\quad (n\neq -1)}$	${\displaystyle {\frac {1}{n+1}}\ln ^{n+1}x}$
${\displaystyle {\frac {1}{x}}\ln {x^{n}}\quad (n\neq 0)}$	${\displaystyle {\frac {1}{2n}}\ln ^{2}{x^{n}}={\frac {n}{2}}\ln ^{2}x}$
${\displaystyle {\frac {1}{x}}{\frac {1}{\ln a}}}$	${\displaystyle \log _{a}x}$
${\displaystyle {\frac {1}{x\ln x}}}$	${\displaystyle \ln \left|\ln x\right|\quad (x>0,x\neq 1)}$
${\displaystyle \log _{a}x}$	${\displaystyle {\frac {1}{\ln a}}(x\ln x-x)}$
${\displaystyle {\sqrt {a^{2}-x^{2}}}}$	${\displaystyle {\frac {a^{2}}{2}}\arcsin \left({\frac {x}{a}}\right)+{\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}}$
${\displaystyle {\sqrt {a^{2}+x^{2}}}}$	${\displaystyle {\frac {a^{2}}{2}}\operatorname {arsinh} \left({\frac {x}{a}}\right)+{\frac {x}{2}}{\sqrt {a^{2}+x^{2}}}}$

Anmerkung:

1. Sonderfall von ${\displaystyle x^{n}}$ für ${\displaystyle n=-1}$, siehe oben in „Potenz- und Wurzelfunktionen“

Trigonometrische Funktionen und Hyperbelfunktionen

Funktion ${\displaystyle f(x)}$	Stammfunktion ${\displaystyle F(x)}$
${\displaystyle \sin x}$	${\displaystyle -\cos x}$
${\displaystyle \cos x}$	${\displaystyle \sin x}$
${\displaystyle \sin ^{2}x}$	${\displaystyle {\tfrac {1}{2}}(x-\sin x\cdot \cos x)}$
${\displaystyle \cos ^{2}x}$	${\displaystyle {\tfrac {1}{2}}(x+\sin x\cdot \cos x)}$
${\displaystyle \sin(kx)\cdot \cos(kx)}$	${\displaystyle -{\frac {1}{4k}}\cos(2kx)}$
${\displaystyle \sin(kx)\cdot \cos(kx)}$	${\displaystyle {\frac {1}{2k}}\sin ^{2}(kx)}$
${\displaystyle \tan x}$	${\displaystyle -\ln |{\cos x}|}$
${\displaystyle \cot x}$	${\displaystyle \ln |{\sin x}|}$
${\displaystyle {\frac {1}{\cos ^{2}x}}=1+\tan ^{2}x}$	${\displaystyle \tan x}$
${\displaystyle {\frac {-1}{\sin ^{2}x}}=-(1+\cot ^{2}x)}$	${\displaystyle \cot x}$
${\displaystyle \arcsin x}$	${\displaystyle x\arcsin x+{\sqrt {1-x^{2}}}}$
${\displaystyle \arccos x}$	${\displaystyle x\arccos x-{\sqrt {1-x^{2}}}}$
${\displaystyle \arctan x}$	${\displaystyle x\arctan x-{\tfrac {1}{2}}\ln \left(1+x^{2}\right)}$
${\displaystyle \operatorname {arccot} x}$	${\displaystyle x\operatorname {arccot} x+{\tfrac {1}{2}}\ln \left(1+x^{2}\right)}$
${\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}}$	${\displaystyle \arcsin x}$
${\displaystyle {\frac {-1}{\sqrt {1-x^{2}}}}}$	${\displaystyle \arccos x}$
${\displaystyle {\frac {1}{x^{2}+1}}}$	${\displaystyle \arctan x}$
${\displaystyle {\frac {x^{2}}{x^{2}+1}}}$	${\displaystyle x-\arctan x}$
${\displaystyle {\frac {1}{(x^{2}+1)^{2}}}}$	${\displaystyle {\frac {1}{2}}\left({\frac {x}{x^{2}+1}}+\arctan x\right)}$
${\displaystyle \sinh x}$	${\displaystyle \cosh x}$
${\displaystyle \cosh x}$	${\displaystyle \sinh x}$
${\displaystyle \tanh x}$	${\displaystyle \ln \cosh x}$
${\displaystyle \coth x}$	${\displaystyle \ln |{\sinh x}|}$
${\displaystyle {\frac {1}{\cosh ^{2}x}}=1-\tanh ^{2}x}$	${\displaystyle \tanh x}$
${\displaystyle {\frac {-1}{\sinh ^{2}x}}=1-\coth ^{2}x}$	${\displaystyle \coth x}$
${\displaystyle \operatorname {arsinh} x}$	${\displaystyle x\operatorname {arsinh} x-{\sqrt {x^{2}+1}}}$
${\displaystyle \operatorname {arcosh} x}$	${\displaystyle x\operatorname {arcosh} x-{\sqrt {x^{2}-1}}}$
${\displaystyle \operatorname {artanh} x}$	${\displaystyle x\operatorname {artanh} x+{\frac {1}{2}}\ln {\left(1-x^{2}\right)}}$
${\displaystyle \operatorname {arcoth} x}$	${\displaystyle x\operatorname {arcoth} x+{\frac {1}{2}}\ln {\left(x^{2}-1\right)}}$
${\displaystyle {\frac {1}{\sqrt {x^{2}+1}}}}$	${\displaystyle \operatorname {arsinh} x}$
${\displaystyle {\frac {1}{\sqrt {x^{2}-1}}}\quad (x>1)}$	${\displaystyle \operatorname {arcosh} x}$
${\displaystyle {\frac {1}{1-x^{2}}}\quad (\left|x\right|<1)}$	${\displaystyle \operatorname {artanh} x}$
${\displaystyle {\frac {1}{1-x^{2}}}\quad (\left|x\right|>1)}$	${\displaystyle \operatorname {arcoth} x}$

Elliptische Funktionen und elliptische Integrale

Funktion ${\displaystyle f(x)}$	Stammfunktion ${\displaystyle F(x)}$
${\displaystyle {\text{sl}}(x)}$	${\displaystyle -\arctan[{\text{cl}}(x)]}$
${\displaystyle {\text{cl}}(x)}$	${\displaystyle \arctan[{\text{sl}}(x)]}$
${\displaystyle {\text{sn}}(x;k)}$	${\displaystyle -{\frac {1}{k}}{\text{artanh}}[k\,{\text{cd}}(x;k)]}$
${\displaystyle {\text{cn}}(x;k)}$	${\displaystyle {\frac {1}{k}}{\text{arcsin}}[k\,{\text{sn}}(x;k)]}$
${\displaystyle {\text{dn}}(x;k)}$	${\displaystyle {\text{am}}(x;k)}$
${\displaystyle {\frac {1}{\sqrt {1-x^{4}}}}}$	${\displaystyle {\text{arcsl}}(x)}$
${\displaystyle {\frac {1}{\sqrt {x^{4}+1}}}}$	${\displaystyle {\sqrt {2}}\,{\text{arcsl}}{\bigl [}x({\sqrt {x^{4}+1}}+1)^{-1/2}{\bigr ]}}$
${\displaystyle {\frac {1}{\sqrt {x^{6}+1}}}}$	${\displaystyle {\frac {1}{6}}{\sqrt[{4}]{27}}F{\biggl [}2\arctan {\biggl (}{\frac {{\sqrt[{4}]{3}}x}{\sqrt {x^{2}+1}}}{\biggr )};\cos {\bigl (}{\frac {\pi }{12}}{\bigr )}{\biggr ]}}$
${\displaystyle {\frac {1-x^{2}}{\sqrt {x^{8}+1}}}}$	${\displaystyle {\frac {1}{2}}\sec {\bigl (}{\frac {\pi }{8}}{\bigr )}F{\bigl \{}\arcsin {\bigl [}{\frac {2\cos(\pi /8)x}{x^{2}+1}}{\bigr ]};\tan {\bigl (}{\frac {\pi }{8}}{\bigr )}{\bigr \}}}$
${\displaystyle {\frac {x^{2}+1}{\sqrt {x^{8}+1}}}}$	${\displaystyle {\frac {1}{2}}\sec {\bigl (}{\frac {\pi }{8}}{\bigr )}F{\biggl \{}2\arctan {\biggl [}{\frac {2\cos(\pi /8)x}{{\sqrt {x^{4}+{\sqrt {2}}x^{2}+1}}-x^{2}+1}}{\biggr ]};2{\sqrt[{4}]{2}}\sin {\bigl (}{\frac {\pi }{8}}{\bigr )}{\biggr \}}}$
${\displaystyle {\frac {1}{\sqrt[{4}]{(ax^{2}+bx+c)^{3}}}}}$	${\displaystyle {\frac {2{\sqrt {2}}}{\sqrt[{4}]{4a^{2}c-ab^{2}}}}\operatorname {arcsl} {\biggl [}{\frac {2ax+b}{{\sqrt {4a(ax^{2}+bx+c)}}+{\sqrt {4ac-b^{2}}}}}{\biggr ]}}$
${\displaystyle {\frac {1}{x(1-x^{2})}}[E(x)-(1-x^{2})K(x)]}$	${\displaystyle K(x)}$
${\displaystyle {\frac {1}{x}}[E(x)-K(x)]}$	${\displaystyle E(x)}$
${\displaystyle K(x)}$	${\displaystyle \int _{0}^{1}{\frac {\arcsin(xy)}{y{\sqrt {1-y^{2}}}}}\mathrm {d} y}$
${\displaystyle E(x)}$	${\displaystyle \int _{0}^{1}{\biggl [}{\frac {\arcsin(xy)}{2y{\sqrt {1-y^{2}}}}}+{\frac {x{\sqrt {1-x^{2}y^{2}}}}{2{\sqrt {1-y^{2}}}}}{\biggr ]}\mathrm {d} y}$
${\displaystyle {\frac {\pi ^{2}q(x)}{2x(1-x^{2})K(x)^{2}}}}$	${\displaystyle q(x)}$
${\displaystyle {\frac {1}{2\pi x}}\vartheta _{10}(x)\vartheta _{00}(x)^{2}E{\biggl [}{\frac {\vartheta _{10}(x)^{2}}{\vartheta _{00}(x)^{2}}}{\biggr ]}}$	${\displaystyle \vartheta _{10}(x)}$
${\displaystyle \vartheta _{00}(x)[\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}]\times }$ ${\displaystyle \times {\biggl \{}{\frac {1}{2\pi x}}E{\biggl [}{\frac {\vartheta _{00}(x)^{2}-\vartheta _{01}(x)^{2}}{\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}}}{\biggr ]}-{\frac {\vartheta _{01}(x)^{2}}{4x}}{\biggr \}}}$	${\displaystyle \vartheta _{00}(x)}$
${\displaystyle -\vartheta _{01}(x)[\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}]\times }$ ${\displaystyle \times {\biggl \{}{\frac {1}{2\pi x}}E{\biggl [}{\frac {\vartheta _{00}(x)^{2}-\vartheta _{01}(x)^{2}}{\vartheta _{00}(x)^{2}+\vartheta _{01}(x)^{2}}}{\biggr ]}-{\frac {\vartheta _{00}(x)^{2}}{4x}}{\biggr \}}}$	${\displaystyle \vartheta _{01}(x)}$

Sonstige

Funktion ${\displaystyle f(x)}$	Stammfunktion ${\displaystyle F(x)}$
${\displaystyle \mathrm {e} ^{-x^{2}}}$	${\displaystyle {\frac {\sqrt {\pi }}{2}}\operatorname {erf} x}$[B 1]
${\displaystyle \mathrm {e} ^{-ax^{2}+bx+c}}$	${\displaystyle {\frac {\sqrt {\pi }}{2{\sqrt {a}}}}\mathrm {e} ^{{\frac {b^{2}}{4a}}+c}\operatorname {erf} \left({\sqrt {a}}x-{\frac {b}{2{\sqrt {a}}}}\right)}$[B 1]
${\displaystyle {\frac {u'(x)}{u(x)}}}$	${\displaystyle \ln \left|u(x)\right|}$
${\displaystyle u'(x)\cdot u(x)}$	${\displaystyle {\tfrac {1}{2}}(u(x))^{2}}$
${\displaystyle u'(x)\cdot (u(x))^{n}}$	${\displaystyle {\frac {1}{n+1}}(u(x))^{n+1}}$

Anmerkung:

1. ${\displaystyle \operatorname {erf} }$ ist die Fehlerfunktion

Rekursionsformeln für weitere Stammfunktionen

• ${\displaystyle \int {\frac {1}{(x^{2}+1)^{n}}}\,\mathrm {d} x={\frac {1}{2n-2}}\cdot {\frac {x}{(x^{2}+1)^{n-1}}}+{\frac {2n-3}{2n-2}}\cdot \int {\frac {1}{(x^{2}+1)^{n-1}}}\,\mathrm {d} x,\quad n\geq 2}$
• ${\displaystyle \int \sin ^{n}(x)\,\mathrm {d} x={\frac {n-1}{n}}\cdot \int \sin ^{n-2}(x)\,\mathrm {d} x-{\frac {1}{n}}\cdot \cos(x)\cdot \sin ^{n-1}(x),\quad n\geq 2}$
• ${\displaystyle \int \cos ^{n}(x)\,\mathrm {d} x={\frac {n-1}{n}}\cdot \int \cos ^{n-2}(x)\,\mathrm {d} x+{\frac {1}{n}}\cdot \sin(x)\cdot \cos ^{n-1}(x),\quad n\geq 2}$

Weblinks

• Online-Tool zum Lösen von Integralen
• Online-Integralrechner von WolframAlpha