Question

Solve
${\int }_0^{\pi /2}\sqrt{1+\cos x}dx$

Answer 1

$I={\int }_0^{\pi /2}\sqrt{1+\cos x}dx$

$={\int }_0^{\pi /2}\sqrt{1+2{\cos }^2\frac{x}{2}-1}$

$=\sqrt{2}{\int }_0^{\pi /2}\cos \frac{x}{2}$

$=\sqrt{2}{\left[2\sin \frac{x}{2}\right]}_0^{\pi /2}$

$=2\sqrt{2}[\sin \frac{\pi }{4}-\sin 0]=2\sqrt{2}\times \frac{1}{\sqrt{2}}=2$