$\int^{\pi}_{0}\frac{\sin^m x}{1+\cos x}dx=2^{m-1} \int^{\pi}_{0}\frac{\sin^m \frac{x}{2}\cos^m \frac{x}{2}}{\cos^2 \frac{x}{2}}dx=2^{m-1} \int^{\pi}_{0}\frac{\sin^m \frac{x}{2}\cos^m \frac{x}{2}}{\cos^2 \frac{x}{2}}dx=2^m \int^{\pi/2}_{0}\frac{\sin^m y\cos^m y}{\cos^2 y}dy=2^{m-1}B(\frac{m+1}{2},\frac{m-1}{2})$
（令$y=x/2$）