Question

The value of $\underset{0}{\overset{\pi /2}{\int }}{\cos }^{2}xdx$ is

• A. $\pi$
• B. $2\pi$
• C. $4\pi$
• D. $\frac{\pi }{4}$

Answer 1

The correct option is D $\frac{\pi }{4}$

Consider $I=\underset{0}{\overset{\pi /2}{\int }}{\cos }^{2}xdx\cdots \left(i\right)$

$\Rightarrow$ Using property, we get
$\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{a}{\overset{b}{\int }}f(a+b-x)dx$
$I=\underset{0}{\overset{\pi /2}{\int }}{\sin }^{2}xdx\cdots \left(ii\right)$

Adding $\left(i\right)$ and $\left(ii\right),$ we get

$\Rightarrow 2I=\underset{0}{\overset{\pi /2}{\int }}1dx$

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

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