# Formelsammlung Mathematik: Bestimmte Integrale: Mehrfachintegrale

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##### 1
${\displaystyle \int _{0}^{1}\int _{0}^{1}(xy)^{xy}\,dx\,dy=\int _{0}^{1}x^{x}\,dx}$

##### 2
${\displaystyle \int _{0}^{1}\int _{0}^{1}{\sqrt {x^{2}+y^{2}}}\,dx\,dy={\frac {1}{3}}\left({\sqrt {2}}+\log \left({\sqrt {2}}+1\right)\right)}$

##### 3
${\displaystyle \int _{0}^{1}\int _{0}^{1}\int _{0}^{1}{\sqrt {x^{2}+y^{2}+z^{2}}}\,dx\,dy\,dz=\log({\sqrt {3}}+1)-{\frac {\log 2}{2}}+{\frac {\sqrt {3}}{4}}-{\frac {\pi }{24}}}$

##### 4
${\displaystyle \int _{0}^{1}\int _{0}^{1}\max \left\{x^{\alpha _{1}-1}\,y^{\beta _{1}-1},x^{\alpha _{2}-1}\,y^{\beta _{2}-1}\right\}\,dx\,dy={\frac {{\frac {\alpha _{1}-\alpha _{2}}{\alpha _{2}}}+{\frac {\beta _{2}-\beta _{1}}{\beta _{1}}}}{\alpha _{1}\beta _{2}-\alpha _{2}\beta _{1}}}\qquad \alpha _{1}>\alpha _{2}>0\,,\,\beta _{2}>\beta _{1}>0}$

##### 5
Ist ${\displaystyle V=\left\{(x_{1},...,x_{n})\geq 0\,\left|\,\left({\frac {x_{1}}{a_{1}}}\right)^{b_{1}}+...+\left({\frac {x_{n}}{a_{n}}}\right)^{b_{n}}\leq 1\right.\right\}}$, so gilt
${\displaystyle \int _{V}x_{1}^{c_{1}-1}\cdots x_{n}^{c_{n}-1}\,dx_{1}\cdots dx_{n}={\frac {{a_{1}}^{c_{1}}\cdots {a_{n}}^{c_{n}}}{b_{1}\cdots b_{n}}}\,{\frac {\Gamma \left({\frac {c_{1}}{b_{1}}}\right)\cdots \Gamma \left({\frac {c_{n}}{b_{n}}}\right)}{\Gamma \left({\frac {c_{1}}{b_{1}}}+...+{\frac {c_{n}}{b_{n}}}+1\right)}}}$

##### 6
${\displaystyle \int _{0}^{1}\cdots \int _{0}^{1}\max \left\{x_{1}^{\alpha _{1}},...,x_{n}^{\alpha _{n}}\right\}\,dx_{1}\cdots dx_{n}={\frac {{\frac {1}{\alpha _{1}}}+...+{\frac {1}{\alpha _{n}}}}{1+{\frac {1}{\alpha _{1}}}+...+{\frac {1}{\alpha _{n}}}}}\qquad \alpha _{1},...,\alpha _{n}>0}$

##### 7
${\displaystyle \int _{0}^{1}\int _{0}^{1}{\frac {(-\log xy)^{s-2}}{1-xy}}\,(1-x)\,dx\,dy=\Gamma (s)\left(\zeta (s)-{\frac {1}{s-1}}\right)\qquad {\text{Re}}(s)>0}$

##### 8
${\displaystyle \int _{0}^{\pi }\int _{0}^{\pi }\int _{0}^{\pi }{\frac {dx\,dy\,dz}{1-\cos x\,\cos y\,\cos z}}={\frac {1}{4}}\left[\Gamma \left({\frac {1}{4}}\right)\right]^{4}}$

##### 9
${\displaystyle \int _{0}^{1}\cdots \int _{0}^{1}\prod _{i=1}^{n}\left(t_{i}^{\alpha -1}\,(1-t_{i})^{\beta -1}\right)\prod _{1\leq i

##### 10.1
${\displaystyle \int _{0}^{\infty }\int _{0}^{\infty }e^{-\left(x+y+{\frac {\lambda ^{3}}{xy}}\right)}\cdot x^{{\frac {1}{3}}-1}\cdot y^{{\frac {2}{3}}-1}\,dx\,dy={\frac {2\pi }{\sqrt {3}}}\,e^{-3\lambda }}$

##### 10.2
${\displaystyle \int _{0}^{\infty }\cdots \int _{0}^{\infty }e^{-\left(x_{1}+x_{2}+...+x_{n-1}+{\frac {z^{n}}{x_{1}\,x_{2}\cdots x_{n-1}}}\right)}\,x_{1}^{{\frac {1}{n}}-1}\,x_{2}^{{\frac {2}{n}}-1}\cdots x_{n-1}^{{\frac {n-1}{n}}-1}\,dx_{1}\,dx_{2}\cdots dx_{n-1}={\frac {1}{\sqrt {n}}}\,{\sqrt {2\pi }}^{\,n-1}\,e^{-nz}}$

##### 11
${\displaystyle \int _{0}^{1}\int _{0}^{1}{\frac {1}{(1-xy)\,{\sqrt {x\,(1-y)}}}}\,dx\,dy=8G}$

##### 12
${\displaystyle \int _{0}^{1}\int _{0}^{1}{\frac {1}{(x+y)\,{\sqrt {(1-x)(1-y)}}}}\,dx\,dy=4G}$

##### 13
${\displaystyle \int _{0}^{1}\int _{0}^{1}{\frac {1}{\sqrt {1+x^{2}+y^{2}}}}\,dx\,dy=\log \left(2+{\sqrt {3}}\right)-{\frac {\pi }{6}}}$

##### 14
${\displaystyle \int _{0}^{1}\int _{0}^{1}{\frac {1}{2-x^{2}-y^{2}}}\,dx\,dy=G}$

##### 15
${\displaystyle \int _{0}^{1}\int _{0}^{1}{\frac {1-x^{2}}{(1+x^{2}y^{2})\,\log ^{2}(xy)}}\,dx\,dy={\frac {2G}{\pi }}-{\frac {1}{2}}-\log \left(2\,\,{\frac {\Gamma \left({\frac {3}{4}}\right)}{\Gamma \left({\frac {1}{4}}\right)}}\right)}$

##### 16
${\displaystyle \int _{0}^{1}\int _{0}^{1}{\frac {x^{\alpha -1}\,y^{\beta -1}}{1+xy}}\,dx\,dy={\frac {1}{2}}\,{\frac {1}{\alpha -\beta }}\left(\left[\psi \left({\frac {1}{2}}+{\frac {\beta }{2}}\right)-\psi \left({\frac {\beta }{2}}\right)\right]-\left[\psi \left({\frac {1}{2}}+{\frac {\alpha }{2}}\right)-\psi \left({\frac {\alpha }{2}}\right)\right]\right)}$

##### 17
${\displaystyle \int _{0}^{1}\int _{0}^{1}{\frac {x^{\alpha -1}\,y^{\beta -1}}{(1+xy)\,\log(xy)}}\,dx\,dy={\frac {1}{\alpha -\beta }}\log {\frac {\Gamma \left({\frac {\alpha }{2}}\right)\,\Gamma \left({\frac {1}{2}}+{\frac {\beta }{2}}\right)}{\Gamma \left({\frac {\beta }{2}}\right)\,\Gamma \left({\frac {1}{2}}+{\frac {\alpha }{2}}\right)}}}$

##### 18
${\displaystyle \int _{0}^{1}\int _{0}^{1}{\frac {x}{(1+x^{2}y^{2})\,\log(xy)}}\,dx\,dy=\log \left({\sqrt {\pi }}\,\,{\frac {\Gamma \left({\frac {3}{4}}\right)}{\Gamma \left({\frac {1}{4}}\right)}}\right)}$

##### 19
${\displaystyle \int _{0}^{1}\int _{0}^{1}{\frac {1-x^{2}}{(1+x^{2}y^{2})\,\log ^{2}(xy)}}\,dx\,dy={\frac {2G}{\pi }}-{\frac {1}{2}}-\log \left(2\,\,{\frac {\Gamma \left({\frac {3}{4}}\right)}{\Gamma \left({\frac {1}{4}}\right)}}\right)}$

##### 20
Ist ${\displaystyle V:=\left\{(x_{1},...,x_{n})\geq 0\,{\Big |}\,x_{1}+...+x_{n}=1\right\}}$ und ist ${\displaystyle 0, so gilt

${\displaystyle \int \limits _{V}{\frac {1}{a_{1}x_{1}+...+a_{n}x_{n}}}\,dx_{1}\cdots dx_{n}={\frac {1}{(n-2)!}}\sum _{k=1}^{n}{\frac {\log a_{k}}{a_{k}}}\,\prod _{j=1 \atop j\neq k}^{n}{\frac {a_{k}}{a_{k}-a_{j}}}}$.