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

Zur Navigation springen Zur Suche springen

Zurück zu Bestimmte Integrale

##### 1.1
${\displaystyle J_{\nu }(z)={\frac {2}{\Gamma \left({\frac {1}{2}}\right)\Gamma \left(\nu +{\frac {1}{2}}\right)}}\left({\frac {z}{2}}\right)^{\nu }\int _{0}^{\frac {\pi }{2}}\cos(z\cos x)\,\sin ^{2\nu }x\,dx\qquad {\text{Re}}(\nu )>-{\frac {1}{2}}}$

##### 1.2
${\displaystyle J_{\nu }(z)={\frac {2}{\Gamma \left({\frac {1}{2}}\right)\Gamma \left(\nu +{\frac {1}{2}}\right)}}\left({\frac {z}{2}}\right)^{\nu }\int _{0}^{\frac {\pi }{2}}\cos(z\sin x)\,\cos ^{2\nu }x\,dx\qquad {\text{Re}}(\nu )>-{\frac {1}{2}}}$

##### 1.3
${\displaystyle \int _{0}^{\frac {\pi }{2}}{\frac {\sin \alpha x}{\sin x}}\,\cos ^{\alpha -1}x\,dx={\frac {\pi }{2}}\qquad {\text{Re}}(\alpha )>0}$

##### 1.4
${\displaystyle \int _{0}^{\frac {\pi }{2}}{\frac {\sin 2\alpha x}{\sin x}}\,\cos x\,dx={\frac {\pi }{2}}+{\frac {\sin \alpha \pi }{2}}\left(\psi \left({\frac {\alpha }{2}}\right)-\psi \left({\frac {\alpha }{2}}+{\frac {1}{2}}\right)+{\frac {1}{\alpha }}\right)}$

##### 1.5
${\displaystyle \int _{0}^{\frac {\pi }{2}}{\frac {k\,\cos \,x}{\sqrt {1-k^{2}\sin ^{2}x}}}\,dx=\arcsin k}$

##### 1.6
${\displaystyle \int _{0}^{\frac {\pi }{2}}{\frac {k\,\sin x\,\cos ^{2}x}{\sqrt {1-k^{2}\sin ^{2}x}}}\,dx={\frac {1}{2k}}+{\frac {k^{2}-1}{2k^{2}}}\,{\text{artanh}}\,k}$

##### 2.1
${\displaystyle \int _{0}^{\pi }\sin nx\,\cos mx\,dx=\left\{{\begin{matrix}0&,&m\equiv n\mod 2\\\\{\frac {1}{n+m}}+{\frac {1}{n-m}}&,&\mathrm {sonst} \end{matrix}}\right.}$

##### 2.2
${\displaystyle \int _{-\pi }^{\pi }\sin ^{n}x\,\cos ^{m}x\,dx={\frac {{\frac {n!}{2^{n}\left({\frac {n}{2}}\right)!}}\,{\frac {m!}{2^{m}\left({\frac {m}{2}}\right)!}}}{\left({\frac {n+m}{2}}\right)!}}}$ wenn ${\displaystyle n,m\in \mathbb {N} }$ beide gerade sind, andernfalls ist das Integral 0.

##### 2.3
${\displaystyle J_{n}(z)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(z\sin x-nx)\,dx\qquad n\in \mathbb {Z} \;,\;z\in \mathbb {C} }$

##### 2.4
${\displaystyle 2\int _{0}^{\frac {\pi }{2}}\sin ^{2\alpha -1}x\,\cos ^{2\beta -1}x\,dx=B(\alpha ,\beta )\qquad {\text{Re}}(\alpha ),{\text{Re}}(\beta )>0}$

##### 3.1
${\displaystyle \int _{0}^{\frac {\pi }{2}}{\frac {dx}{(a^{2}\,\cos ^{2}x+b^{2}\,\sin ^{2}x)^{n+1}}}={\frac {\pi }{2ab}}\,\sum _{k=0}^{n}{\frac {2k \choose k}{(2a)^{2k}}}\,{\frac {2(n-k) \choose n-k}{(2b)^{2(n-k)}}}}$