Man berechne:
−∫dxa2−x2=arcosxa{\displaystyle -\int {\frac {dx}{\sqrt {a^{2}-x^{2}}}}=arcos{\frac {x}{a}}}
∫x∗arcsin(x)dx=x22arcsin(x)−12∫x22dx1−x2{\displaystyle \int x*arcsin(x)\,dx={\frac {x^{2}}{2}}arcsin(x)-{\frac {1}{2}}\int {\frac {x^{2}}{2}}{\frac {dx}{\sqrt {1-x^{2}}}}}
Partielle Integration
∫x∗arcsin(x)dx=x22arcsin(x)+12arcos(x)x2−∫arcos(x)2xdx{\displaystyle \int x*arcsin(x)\,dx={\frac {x^{2}}{2}}arcsin(x)+{\frac {1}{2}}arcos(x)x^{2}-\int arcos(x)2x\,dx}
Umformen des arcos(x)
∫x∗arcsin(x)dx=x22arcsin(x)+x22(π2−arcsin(x))−∫(π22x−arcsin(x)2x)dx{\displaystyle \int x*arcsin(x)\,dx={\frac {x^{2}}{2}}arcsin(x)+{\frac {x^{2}}{2}}({\frac {\pi }{2}}-arcsin(x))-\int ({\frac {\pi }{2}}2x-arcsin(x)2x)\,dx}
Teilen des Integrals und Integration des ersten Teiles
∫x∗arcsin(x)dx=x22arcsin(x)+πx24−x22arcsin(x))−πx2x+2∫x∗arcsin(x)dx{\displaystyle \int x*arcsin(x)\,dx={\frac {x^{2}}{2}}arcsin(x)+{\frac {\pi x^{2}}{4}}-{\frac {x^{2}}{2}}arcsin(x))-{\frac {\pi x^{2}}{x}}+2\int x*arcsin(x)\,dx}
Nun kann man die arcsin abziehen
−∫x∗arcsin(x)dx=+πx24−πx{\displaystyle -\int x*arcsin(x)\,dx=+{\frac {\pi x^{2}}{4}}-\pi x}
Mein Endergebnis:
∫x∗arcsin(x)dx=πx−πx24{\displaystyle \int x*arcsin(x)\,dx=\pi x-{\frac {\pi x^{2}}{4}}}
