Question

Solve: ${\int }_0^{\pi /2}{\cos }^{2}xdx$ = $\frac{\pi }{k}$ then k =

Answer 1

${\int }_0^{\frac{\pi }{2}}{\cos }^2\left(x\right)dx$

$={\int }_0^{\frac{\pi }{2}}\frac{1+\cos \left(2x\right)}{2}dx$

$=\frac{1}{2}\cdot {\int }_0^{\frac{\pi }{2}}1+\cos \left(2x\right)dx$

$=\frac{1}{2}({\int }_0^{\frac{\pi }{2}}1dx+{\int }_0^{\frac{\pi }{2}}\cos \left(2x\right)dx)$

$=\frac{1}{2}(\frac{\pi }{2}+0)$

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