Question

The value of $\underset{0}{\overset{\pi }{\int }}|cosx{|}^{3}dx$ is equal to $\frac{k}{3}$, then the value of $k$ is

Answer 1

Answer: 4

$I=\underset{0}{\overset{\pi }{\int }}|cosx{|}^{3}dx=2\underset{0}{\overset{\pi /2}{\int }}co{s}^{2}xdx$
$=\frac{2}{4}\underset{0}{\int }(3cosx+cos3x)dx$
$[\because cos3\theta =4co{s}^3\theta -3cos\theta ]$
$=\frac{1}{2}[3sinx+\frac{sin3x}{3}{]}_0^{\pi /2}$
$=\frac{1}{2}(3-\frac{1}{3})=\frac{4}{3}$
$\Rightarrow \frac{k}{3}=\frac{4}{3}$
$\Rightarrow k=4$