Question

Solve:${\int }_0^{\pi }\sqrt{1-\sin x}dx$

Answer 1

$1-\sin x={\cos }^2\frac{x}{2}+{\sin }^2\frac{x}{2}-2\sin \frac{x}{2}\cos \frac{x}{2}=(\cos \frac{x}{2}-\sin \frac{x}{2}{)}^2$

$⟹{\int }_0^{\pi }(\cos \frac{x}{2}-\sin \frac{x}{2})dx$

$⟹\left[\frac{1}{2}\right(-\sin \frac{x}{2}-\cos \frac{x}{2}){]}_0^{\pi }=0$