Question

Evaluate :

$\underset{0}{\overset{\pi /2}{\int }}{\cos }^{2}xdx$

Answer 1

Consider $I=\underset{0}{\overset{\pi /2}{\int }}{\cos }^{2}xdx$ .........(1)

$I=\underset{0}{\overset{\pi /2}{\int }}{\cos }^2(\frac{\pi }{2}-x)dx$ $\left(\underset{0}{\overset{a}{\int }}f\right(x)dx=\underset{0}{\overset{a}{\int }}f(a-x\left)dx\right)$

$I=\underset{0}{\overset{\pi /2}{\int }}{\sin }^{2}xdx$ .....(2)

Adding 1 and 2, we get,

$2I=\underset{0}{\overset{\pi /2}{\int }}dx$

$2I=\frac{\pi }{2}-0$

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