Formelsammlung Mathematik: Bestimmte Integrale: Mehrfachintegrale

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1

$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}(xy{)}^{xy}dxdy=\underset{0}{\overset{1}{\int }}{x}^{x}dx$

2

$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\sqrt{x^2+y^2}dxdy=\frac{1}{3}(\sqrt{2}+\log (\sqrt{2}+1))$

3

$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\sqrt{x^2+y^2+z^2}dxdydz=\log (\sqrt{3}+1)-\frac{\log 2}{2}+\frac{\sqrt{3}}{4}-\frac{\pi }{24}$

4

$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\max \{x^{{\alpha }_1-1}{y}^{{\beta }_1-1},x^{{\alpha }_2-1}{y}^{{\beta }_2-1}\}dxdy=\frac{\frac{{\alpha }_1-{\alpha }_2}{{\alpha }_2}+\frac{{\beta }_2-{\beta }_1}{{\beta }_1}}{{\alpha }_1{\beta }_2-{\alpha }_2{\beta }_1}{\alpha }_1>{\alpha }_2>0,{\beta }_2>{\beta }_1>0$

5

Ist $V=\{(x_1,...,x_n)\ge 0|{\left(\frac{x_1}{a_1}\right)}^{b_1}+...+{\left(\frac{x_n}{a_n}\right)}^{b_n}\le 1\}$, so gilt

$\underset{V}{\int }{x}_1^{c_1-1}\cdots x_n^{c_n-1}d{x}_1\cdots d{x}_n=\frac{{a_1}^{c_1}\cdots {a_n}^{c_n}}{b_1\cdots b_n}\frac{\mathrm{\Gamma }\left(\frac{c_1}{b_1}\right)\cdots \mathrm{\Gamma }\left(\frac{c_n}{b_n}\right)}{\mathrm{\Gamma }(\frac{c_1}{b_1}+...+\frac{c_n}{b_n}+1)}$

6

$\underset{0}{\overset{1}{\int }}\cdots \underset{0}{\overset{1}{\int }}\max \{x_1^{{\alpha }_1},...,x_n^{{\alpha }_n}\}d{x}_1\cdots d{x}_n=\frac{\frac{1}{{\alpha }_1}+...+\frac{1}{{\alpha }_n}}{1+\frac{1}{{\alpha }_1}+...+\frac{1}{{\alpha }_n}}{\alpha }_1,...,{\alpha }_n>0$

7

$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\frac{(-\log xy{)}^{s-2}}{1-xy}(1-x)dxdy=\mathrm{\Gamma }(s)(\zeta (s)-\frac{1}{s-1})\text{Re}(s)>0$

8

$\underset{0}{\overset{\pi }{\int }}\underset{0}{\overset{\pi }{\int }}\underset{0}{\overset{\pi }{\int }}\frac{dxdydz}{1-\cos x\cos y\cos z}=\frac{1}{4}{\left[\mathrm{\Gamma }\left(\frac{1}{4}\right)\right]}^4$

9

$\underset{0}{\overset{1}{\int }}\cdots \underset{0}{\overset{1}{\int }}\prod _{i=1}^n(t_i^{\alpha -1}(1-t_i{)}^{\beta -1})\prod _{1\le i<j\le n}|t_i-t_j{|}^{2\gamma }d{t}_1\cdots d{t}_n=\prod _{j=0}^{n-1}\frac{\mathrm{\Gamma }(\alpha +j\gamma )\mathrm{\Gamma }(\beta +j\gamma )\mathrm{\Gamma }(1+(j+1)\gamma )}{\mathrm{\Gamma }(\alpha +\beta +(n+j-1)\gamma )\mathrm{\Gamma }(1+\gamma )}$

10.1

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-(x+y+\frac{{\lambda }^3}{xy})}\cdot x^{\frac{1}{3}-1}\cdot y^{\frac{2}{3}-1}dxdy=\frac{2\pi }{\sqrt{3}}{e}^{-3\lambda }$

10.2

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\cdots \underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-(x_1+x_2+...+x_{n-1}+\frac{z^n}{x_1{x}_2\cdots x_{n-1}})}{x}_1^{\frac{1}{n}-1}{x}_2^{\frac{2}{n}-1}\cdots x_{n-1}^{\frac{n-1}{n}-1}d{x}_{1}d{x}_2\cdots d{x}_{n-1}=\frac{1}{\sqrt{n}}{\sqrt{2\pi }}^{n-1}{e}^{-nz}$

11

$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\frac{1}{(1-xy)\sqrt{x(1-y)}}dxdy=8G$

12

$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\frac{1}{(x+y)\sqrt{(1-x)(1-y)}}dxdy=4G$

13

$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\frac{1}{\sqrt{1+x^2+y^2}}dxdy=\log (2+\sqrt{3})-\frac{\pi }{6}$

14

$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\frac{1}{2-x^2-y^2}dxdy=G$

15

$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\frac{1-x^2}{(1+x^2{y}^2){\log }^2(xy)}dxdy=\frac{2G}{\pi }-\frac{1}{2}-\log \left(2\frac{\mathrm{\Gamma }\left(\frac{3}{4}\right)}{\mathrm{\Gamma }\left(\frac{1}{4}\right)}\right)$

16

$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\frac{x^{\alpha -1}{y}^{\beta -1}}{1+xy}dxdy=\frac{1}{2}\frac{1}{\alpha -\beta }([\psi (\frac{1}{2}+\frac{\beta }{2})-\psi \left(\frac{\beta }{2}\right)]-[\psi (\frac{1}{2}+\frac{\alpha }{2})-\psi \left(\frac{\alpha }{2}\right)])$

17

$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\frac{x^{\alpha -1}{y}^{\beta -1}}{(1+xy)\log (xy)}dxdy=\frac{1}{\alpha -\beta }\log \frac{\mathrm{\Gamma }\left(\frac{\alpha }{2}\right)\mathrm{\Gamma }(\frac{1}{2}+\frac{\beta }{2})}{\mathrm{\Gamma }\left(\frac{\beta }{2}\right)\mathrm{\Gamma }(\frac{1}{2}+\frac{\alpha }{2})}$

18

$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\frac{x}{(1+x^2{y}^2)\log (xy)}dxdy=\log \left(\sqrt{\pi }\frac{\mathrm{\Gamma }\left(\frac{3}{4}\right)}{\mathrm{\Gamma }\left(\frac{1}{4}\right)}\right)$

19

$\underset{0}{\overset{1}{\int }}\underset{0}{\overset{1}{\int }}\frac{1-x^2}{(1+x^2{y}^2){\log }^2(xy)}dxdy=\frac{2G}{\pi }-\frac{1}{2}-\log \left(2\frac{\mathrm{\Gamma }\left(\frac{3}{4}\right)}{\mathrm{\Gamma }\left(\frac{1}{4}\right)}\right)$

20

Ist $V:=\{(x_1,...,x_n)\ge 0|x_1+...+x_n=1\}$ und ist $0<a_1<...<a_n$, so gilt

$\underset{V}{\int }\frac{1}{a_1{x}_1+...+a_n{x}_n}d{x}_1\cdots d{x}_n=\frac{1}{(n-2)!}\sum _{k=1}^n\frac{\log a_k}{a_k}\prod _{\genfrac{}{}{0ex}{}{j=1}{j\ne k}}^n\frac{a_k}{a_k-a_j}$.