Question

$\underset{0}{\overset{\mathrm{\pi }/2}{\int }}\frac{\sin x}{\sqrt{1+\cos x}}dx$

Answer 1

$$\begin{gathered} {\int }_0^{\frac{\mathrm{\pi }}{2}}\frac{\sin x}{\sqrt{1+\cos x}}dx \\ \mathrm{Let}1+\cos x=t,\mathrm{then}-\sin xdx=dt \\ \mathrm{When},x\to 0,t\to 2\mathrm{and}x\to \frac{\mathrm{\pi }}{2},t\to 1 \\ \mathrm{Therefore},\mathrm{the}\mathrm{integral}\mathrm{becomes} \\ {\int }_2^1\frac{-1}{\sqrt{t}}dt \\ ={\int }_1^2\frac{1}{\sqrt{t}}dt \\ =2{\left[\sqrt{t}\right]}_1^2 \\ =2\left(\sqrt{2}-1\right) \end{gathered}$$