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

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1.1

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

1.2

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

1.3

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\sin \alpha x}{\sin x}{\cos }^{\alpha -1}xdx=\frac{\pi }{2}\text{Re}(\alpha )>0$

1.4

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{\sin 2\alpha x}{\sin x}\cos xdx=\frac{\pi }{2}+\frac{\sin \alpha \pi }{2}(\psi \left(\frac{\alpha }{2}\right)-\psi (\frac{\alpha }{2}+\frac{1}{2})+\frac{1}{\alpha })$

1.5

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{k\cos x}{\sqrt{1-k^2{\sin }^{2}x}}dx=\arcsin k$

1.6

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{k\sin x{\cos }^{2}x}{\sqrt{1-k^2{\sin }^{2}x}}dx=\frac{1}{2k}+\frac{k^2-1}{2k^2}\text{artanh}k$

2.1

$\underset{0}{\overset{\pi }{\int }}\sin nx\cos mxdx=\{\begin{array}{ccc}0& ,& m\equiv nmod2\\ \\ \frac{1}{n+m}+\frac{1}{n-m}& ,& \mathrm{s}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\end{array}$

2.2

$\underset{-\pi }{\overset{\pi }{\int }}{\sin }^{n}x{\cos }^{m}xdx=\frac{\frac{n!}{2^n\left(\frac{n}{2}\right)!}\frac{m!}{2^m\left(\frac{m}{2}\right)!}}{\left(\frac{n+m}{2}\right)!}$ wenn $n,m\in \mathbb{N}$ beide gerade sind, andernfalls ist das Integral 0.

2.3

$J_n(z)=\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}\cos (z\sin x-nx)dxn\in \mathbb{Z},z\in ℂ$

2.4

$2\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\sin }^{2\alpha -1}x{\cos }^{2\beta -1}xdx=B(\alpha ,\beta )\text{Re}(\alpha ),\text{Re}(\beta )>0$

3.1

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{dx}{(a^2{\cos }^{2}x+b^2{\sin }^{2}x{)}^{n+1}}=\frac{\pi }{2ab}\sum _{k=0}^n\frac{(\genfrac{}{}{0ex}{}{2k}{k})}{(2a{)}^{2k}}\frac{(\genfrac{}{}{0ex}{}{2(n-k)}{n-k})}{(2b{)}^{2(n-k)}}$