Question

Evaluate : ${\int }_0^{\pi /2}{\cos }^{2}xdx$

Answer 1

${\int }_0^{\pi /2}{\cos }^{2}xdx$

$={\int }_0^{\pi /2}\frac{\cos 2\theta +1}{2}dx$

$=\frac{1}{2}{[\frac{\sin 2\theta }{2}+\theta ]}_0^{\pi /2}$

$=\frac{1}{2}[\frac{\sin 2\left(\pi /2\right)}{2}+\frac{\pi }{2}+\frac{\sin 2\left(0\right)}{2}+0]$

$=\frac{1}{2}\left[\frac{\pi }{2}\right]$

$=\frac{\pi }{4}$