Question

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

Answer 1

${\int }_0^{\pi /2}co{s}^{2}xdx={\int }_0^{\pi /2}co{s}^2(\frac{\pi }{2}-x)dx=I$

$\therefore I+I={\int }_0^{\pi /2}co{s}^{2}xdx+{\int }_0^{\pi /2}si{n}^{2}xdx$

$\therefore 2I={\int }_0^{\pi /2}1dx$

$\therefore 2I=\frac{\pi }{2}$

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