Question Number 156339 by lapache last updated on 10/Oct/21
$$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{\mathrm{2}{n}} }{{arcsinx}}{dx}=…??? \\ $$
Answered by ArielVyny last updated on 10/Oct/21
$${t}={arcsinx}\rightarrow{sint}={x} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{sin}^{\mathrm{2}{n}} {t}}{{t}}{costdt}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{sin}^{\mathrm{2}{n}} {t}}{{t}}\underset{{n}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{{n}} {t}^{\mathrm{2}{n}} }{\left(\mathrm{2}{n}\right)!} \\ $$$$\underset{{n}\geqslant\mathrm{0}} {\sum}\frac{\left(−\mathrm{1}\right)^{{n}} }{\left(\mathrm{2}{n}\right)!}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {t}^{\mathrm{2}{n}} {sin}^{\mathrm{2}{n}} {tdt} \\ $$