# $I= \int \limits_0^{\frac{\pi}{2}} \cos ^3 x dx$

$\begin {array} {1 1} (a)\;\frac{1}{3} \\ (b)\;1 \\ (c)\;\frac{2}{3} \\ (d)\;0 \end {array}$

$I= \int \limits_0^{\frac{\pi}{2}} \cos ^3 x dx$
$\quad= \int \limits_0^{\frac{\pi}{2}} \cos x (\cos ^2 x) dx$
$\quad= \int \limits_0^{\frac{\pi}{2}} \cos x (1-\sin ^2 x) dx$
$\quad= \int \limits_0^{\frac{\pi}{2}} \cos x dx -\int \limits_0^{\frac{\pi}{2}} \cos x \sin ^2 x dx$
$\sin x =t \quad \cos x dx=dt$
$\qquad= \sin x \bigg]_0^{\pi/2} - \int \limits _0^1 t^2 dt$
$\qquad=1-\large\frac{1}{3}$
$\qquad=\large\frac{2}{3}$
Hence c is the correct answer.