Question

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

Answer 1

Answer: 3

$\underset{0}{\overset{x}{\int }}{\left|cosx\right|}^{3}dx=2\underset{0}{\overset{\frac{\pi }{2}}{\int }}co{s}^{3}xdx$
$=2\underset{0}{\overset{\frac{\pi }{2}}{\int }}\left(\frac{cos3x+3cosx}{4}\right)dx$
$=\frac{1}{2}{\left[\frac{sin3x}{3}+3sinx\right]}_0^{\frac{\pi }{2}}$
$=\frac{1}{2}\left(-\frac{1}{3}+3\right)$
$=\frac{4}{3}$ sq.units
$=\frac{3+1}{3}$
$\Rightarrow k=3$