Formelsammlung Mathematik: Bestimmte Integrale: Form R(x,cos)

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1.1

$\frac{1}{2\pi i}\underset{n-i\mathrm{\infty }}{\overset{n+i\mathrm{\infty }}{\int }}\frac{\pi }{x\cos \pi x}dx=2\sum _{k=n}^{\mathrm{\infty }}\frac{(-1{)}^k}{2k+1}n\in {\mathbb{Z}}^{>0}$

1.2

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\cos \alpha x}{1+x^2}dx=\pi e^{-\alpha }\alpha \ge 0$

1.3

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\cos \alpha x}{1-x^2}dx=\pi \sin \alpha \alpha \ge 0$

1.4

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{|\cos \alpha x|}{1+x^2}dx=4\cosh \alpha \arctan e^{-\alpha }\alpha >0$

1.5

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\cos (x^2-\frac{{\alpha }^2}{x^2})dx=\sqrt{\frac{\pi }{2}}{e}^{-2\alpha }$

1.6

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\cos (x^2+\frac{{\alpha }^2}{x^2})dx=\sqrt{\frac{\pi }{2}}(\cos 2\alpha -\sin 2\alpha )$

1.7

$J_{\nu }(z)=\frac{2}{\mathrm{\Gamma }\left(\frac{1}{2}\right)\mathrm{\Gamma }(\nu +\frac{1}{2})}{\left(\frac{z}{2}\right)}^{\nu }\underset{0}{\overset{1}{\int }}\cos (zx)(1-x^2{)}^{\nu -\frac{1}{2}}dx\text{Re}(\nu )>-\frac{1}{2}$

1.8

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\cos }^{2n}(x)dx=\frac{1}{2^{2n}}(\genfrac{}{}{0ex}{}{2n}{n})\frac{\pi }{2}n\in \mathbb{N}$

1.9

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\cos }^{2n+1}(x)dx=\frac{1}{2n+1}{\left[\frac{1}{2^{2n}}(\genfrac{}{}{0ex}{}{2n}{n})\right]}^{-1}n\in \mathbb{N}$

2.1

$\underset{0}{\overset{\pi }{\int }}\cos nx\cos mxdx={\delta }_{mn}\frac{\pi }{2}n,m\in {\mathbb{Z}}^{\ge 1}$

2.2

$\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}{x}^2\cos nx\cos mxdx=\{\begin{array}{ccc}\frac{{\pi }^2}{6}\pm \frac{1}{4nm}& ,& n=m\\ \\ \frac{(-1{)}^{n-m}}{(n-m{)}^2}\pm \frac{(-1{)}^{n+m}}{(n+m{)}^2}& ,& n\ne m\end{array}n,m\in {\mathbb{Z}}^{>0}$

2.3

$\underset{0}{\overset{\pi }{\int }}\frac{\cos nx-\cos na}{\cos x-\cos a}dx=\pi \frac{\sin na}{\sin a}n\in \mathbb{N},a\in ℂ$

2.4

$\underset{-\pi }{\overset{\pi }{\int }}\frac{\cos nx}{1-2r\cos x+r^2}dx=\frac{2\pi r^n}{1-r^2}|r|<1,n\in {\mathbb{Z}}^{\ge 0}$

2.5

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{|\cos \alpha x|-|\cos \beta x|}{x}dx=(1-\frac{2}{\pi })\log \frac{\beta }{\alpha }\alpha ,\beta >0$

2.6

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{|\cos \alpha x|-|\cos \beta x|}{x^2}dx=\beta -\alpha \alpha ,\beta >0$

2.7

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\cos (\alpha x)}{1+2\cos \theta x+x^2}dx=\frac{\pi }{\sin \theta }\frac{\cos (\alpha \cos \theta )}{e^{\alpha \sin \theta }}\alpha \ge 0,\theta \in ℂ\setminus \pi \mathbb{Z}$

2.8

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\cos (\alpha x)}{x^4+{\beta }^4}dx=\frac{\pi }{{\beta }^3\sqrt{2}}(\cos \frac{\alpha \beta }{\sqrt{2}}+\sin \frac{\alpha \beta }{\sqrt{2}})e^{-\frac{\alpha \beta }{\sqrt{2}}}\alpha ,\beta >0$

2.9

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\cos \alpha x}{\prod _{k=0}^{\mathrm{\infty }}(1+\frac{x^2}{(\beta +k{)}^2})}dx=\sqrt{\pi }\frac{\mathrm{\Gamma }(\beta +\frac{1}{2})}{\mathrm{\Gamma }(\beta )}{\text{sech}}^{2\beta }\left(\frac{\alpha }{2}\right)\alpha ,\beta >0$

2.10

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\cos }^{\alpha -1}x\cos \beta xdx=\frac{\pi }{2^{\alpha }}\frac{\mathrm{\Gamma }(\alpha )}{\mathrm{\Gamma }\left(\frac{\alpha +\beta +1}{2}\right)\mathrm{\Gamma }\left(\frac{\alpha -\beta +1}{2}\right)}\text{Re}(\alpha )>0,\beta \in ℂ$

3.1

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{\cos \alpha x}{({\beta }^2+x^2)((\beta +1{)}^2+x^2)\cdots ((\beta +n{)}^2+x^2)}dx=2\pi \sum _{k=0}^n(-1{)}^k\frac{(2\beta -1+k)!}{(2\beta +n+k)!}\frac{1}{k!(n-k)!}{e}^{-\alpha (\beta +k)}n\in \mathbb{N},\alpha ,\beta >0$

3.2

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\cos (\alpha t^{\frac{1}{z}}+\beta )dt=\frac{\mathrm{\Gamma }(z+1)}{{\alpha }^z}\cos (\frac{\pi z}{2}+\beta )0<z<1,\alpha >0,\beta \in ℂ$