|||
$\int^{\pi}_{0}\frac{\sin^m x}{1+\cos x}dx=2^{m-1} \int^{\pi}_{0}\frac{\sin^m \frac{x}{2}\cos^m \frac{x}{2}}{\cos^2 \frac{x}{2}}dx=2^{m-1} \int^{\pi}_{0}\frac{\sin^m \frac{x}{2}\cos^m \frac{x}{2}}{\cos^2 \frac{x}{2}}dx=2^m \int^{\pi/2}_{0}\frac{\sin^m y\cos^m y}{\cos^2 y}dy=2^{m-1}B(\frac{m+1}{2},\frac{m-1}{2})$
(令$y=x/2$)
Archiver|手机版|科学网 ( 京ICP备07017567号-12 )
GMT+8, 2024-5-17 15:20
Powered by ScienceNet.cn
Copyright © 2007- 中国科学报社