Question

The value of $\int\limits_0^{\pi} |cos\,x|^3 \,dx$ is equal to $\frac{k}{3}$, then the value of $k$ is

Answer 1

$I = \int\limits_{0}^{\pi} |cos\,x|^3 dx = 2\int\limits_0^{\pi/2} cos^2 \,x \,dx$
$ = \frac{2}{4} \int\limits_{0} (3\,cos\,x + cos\,3x) dx$
$[\because cos\,3\theta = 4\, cos^3 \,\theta - 3\,cos\,\theta]$
$ = \frac{1}{2} [3\,sin \,x + \frac{sin\,3x}{3}]_0^{\pi/2}$
$ = \frac{1}{2} ( 3 - \frac{1}{3}) = \frac{4}{3}$
$\Rightarrow \frac{k}{3} = \frac{4}{3}$
$ \Rightarrow k = 4$