Formelsammlung Mathematik: Bestimmte Integrale: Form R(x)

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1.1

$\underset{0}{\overset{1}{\int }}\frac{1-x^{z-1}}{1-x}dx=\gamma +\psi (z)\text{Re}(z)>0$

1.2

$\underset{0}{\overset{1}{\int }}\frac{1-x^{\alpha -1}}{{\sqrt{1-x^2}}^3}dx=2^{\alpha -2}\frac{{\mathrm{\Gamma }}^2\left(\frac{\alpha }{2}\right)}{\mathrm{\Gamma }(\alpha -1)}$

1.3

$\underset{0}{\overset{1}{\int }}x\cdot \sqrt{\frac{1-{\alpha }^2{x}^2}{1-x^2}}dx=\frac{1}{2}+\frac{1-{\alpha }^2}{2\alpha }\text{artanh}\alpha$

1.4

$\underset{0}{\overset{1}{\int }}\frac{1}{\sqrt{(1-x^2)(1-{\alpha }^2{x}^2)}}\mathrm{d}x=K(\alpha )$

1.5

$\underset{0}{\overset{1}{\int }}\sqrt{\frac{1-{\alpha }^2{x}^2}{1-x^2}}\mathrm{d}x=E(\alpha )$

1.6

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\left\{\frac{1}{x}\right\}{x}^{s-1}dx=-\frac{\zeta (s)}{s}0<\text{Re}(s)<1$

2.1

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^{m-1}}{1+x+...+x^{n-1}}dx=\frac{\pi }{n}[\cot \left(\frac{m\pi }{n}\right)-\cot \left(\frac{(m+1)\pi }{n}\right)]0<m<n-1$

2.2

$B(\alpha ,\beta )=\underset{0}{\overset{1}{\int }}{x}^{\alpha -1}(1-x{)}^{\beta -1}dx\text{Re}(\alpha ),\text{Re}(\beta )>0$

2.3

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^{\alpha -1}}{(1+x{)}^{\alpha +\beta }}dx=B(\alpha ,\beta )\text{Re}(\alpha ),\text{Re}(\beta )>0$

2.4

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^{\alpha -1}}{1+x^{\beta }}dx=\frac{\pi }{\beta }\csc \left(\frac{\alpha \pi }{\beta }\right)0<\mathrm{Re}(\alpha )<\mathrm{Re}(\beta )$

2.5

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^{\alpha -1}}{1-x^{\beta }}dx=\frac{\pi }{\beta }\cot \left(\frac{\alpha \pi }{\beta }\right)0<\text{Re}(\alpha )<\text{Re}(\beta )$

2.6

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

2.7

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^{\alpha }}{x^2+2\cos \theta \cdot x+1}dx=\frac{\pi }{\sin \alpha \pi }\frac{\sin \alpha \theta }{\sin \theta }-1<\text{Re}(\alpha )<1,-\pi <\text{Re}(\theta )<\pi$

2.8

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{1\text{oder}{x}^2}{(x^2+2x\cos \alpha +1)(x^2+2x\cos \beta +1)}dx=\frac{1}{2}\frac{\alpha \cot \alpha -\beta \cot \beta }{\cos \alpha -\cos \beta }$

2.9

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x}{(x^2+2x\cos \alpha +1)(x^2+2x\cos \beta +1)}dx=-\frac{1}{2}\frac{\alpha \csc \alpha -\beta \csc \beta }{\cos \alpha -\cos \beta }$

2.10

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{1}{\sqrt{[x^2+2x\sin (\alpha )+1][x^2+2x\sin (\beta )+1]}}dx=2\sec (\frac{\alpha }{2}+\frac{\beta }{2})K[\sin (\frac{\alpha }{2}-\frac{\beta }{2})\sec (\frac{\alpha }{2}+\frac{\beta }{2}\left)\right]-\pi <\alpha +\beta <\pi$

2.11

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{1}{1+\frac{x^2}{a^2}}\prod _{k=1}^{\mathrm{\infty }}\frac{1+\frac{x^2}{(b+k{)}^2}}{1+\frac{x^2}{(a+k{)}^2}}dx=\sqrt{\pi }\frac{\mathrm{\Gamma }(a+\frac{1}{2})\cdot \mathrm{\Gamma }(b+1)\cdot \mathrm{\Gamma }(b-a+\frac{1}{2})}{\mathrm{\Gamma }(a)\cdot \mathrm{\Gamma }(b+\frac{1}{2})\cdot \mathrm{\Gamma }(b-a+1)}0<a<b+\frac{1}{2}$

2.12

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^{2m-1}}{\prod _{k=1}^n(x^2+k^2)}dx=2\cdot (-1{)}^{m-1}\sum _{k=1}^n\frac{(-1{)}^k{k}^{2m}\log k}{(n+k)!(n-k)!}m<n$

2.13

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^{2m-1}}{\prod _{k=0}^n(x^2+(2k+1{)}^2)}dx=\sum _{k=0}^n\frac{(-1{)}^{m+k}(2k+1{)}^{2m-1}\log (2k+1)}{2^{2n}(n+k+1)!(n-k)!}m\le n$

2.14

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^{2m}}{\prod _{k=1}^n(x^2+k^2)}dx=\pi \cdot \sum _{k=1}^n\frac{(-1{)}^{m-1+k}{k}^{2m+1}}{(n+k)!(n-k)!}m<n$

2.15

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^{2m}}{\prod _{k=0}^n(x^2+(2k+1{)}^2)}dx=\pi \cdot \sum _{k=0}^n\frac{(-1{)}^{m+k}(2k+1{)}^{2m}}{2^{2n+1}(n+k+1)!(n-k)!}m\le n$

2.16

$\underset{0}{\overset{\mathrm{\infty }}{\int }}{\left(\frac{2}{1+\sqrt{1+4x}}\right)}^p\cdot x^{s-1}dx=\frac{p\cdot \mathrm{\Gamma }(s)\cdot \mathrm{\Gamma }(p-2s)}{\mathrm{\Gamma }(p-s+1)}0<\text{Re}(s)<\frac{1}{2}\cdot \text{Re}(p)$

3.1

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^{\alpha -1}}{(w+x{)}^{\beta }}dx=\frac{B(\alpha ,\beta -\alpha )}{w^{\beta -\alpha }}0<\text{Re}(\alpha )<\text{Re}(\beta ),\text{Re}(w)>0$

3.2

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{x^{\alpha -1}}{(1+x^{\beta }{)}^{\gamma }}dx=\frac{1}{\beta }B(\frac{\alpha }{\beta },\gamma -\frac{\alpha }{\beta })$

3.3

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\prod _{1\le \ell \le 3}\frac{1}{(x^2+2x\cos {\alpha }_{\ell }+1)}dx=\frac{1}{4}\sum _{(i,j,k)\in A_3}\frac{{\alpha }_i\csc {\alpha }_i\cos 2{\alpha }_i}{(\cos {\alpha }_i-\cos {\alpha }_j)(\cos {\alpha }_i-\cos {\alpha }_k)}$

4.1

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{dx}{(u+ix{)}^{\alpha }(v-ix{)}^{\beta }}=\frac{2\pi }{(u+v{)}^{\alpha +\beta -1}}\frac{\mathrm{\Gamma }(\alpha +\beta -1)}{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(\beta )}\text{Re}(u),\text{Re}(v)>0,\text{Re}(\alpha +\beta )>1$