Question

If $\int\limits_{0}^{\pi }\left|cos x\right|^{3}dx=\frac{k + 1}{k}$ , where $k>0$ , then the value of $k$ is equal to

Answer 1

$\int \limits_{0}^{x}\left|cos x\right|^{3}dx=2\int\limits_{0}^{\frac{\pi }{2}}cos^{3}xdx$
$=2\int \limits_{0}^{\frac{\pi }{2}}\left(\frac{cos 3 x + 3 cos x}{4}\right)dx$
$=\frac{1}{2}\left[\frac{sin 3 x}{3} + 3 sin x\right]_{0}^{\frac{\pi }{2}}$
$=\frac{1}{2}\left(- \frac{1}{3} + 3\right)$
$=\frac{4}{3}$ sq.units
$=\frac{3 + 1}{3}$
$\Rightarrow k=3$