Question

The value of $\underset{0}{\overset{\pi }{\int }}|\cos x{|}^{3}dx$ is :

• A. $0$
• B. $\frac{2}{3}$
• C. $\frac{4}{3}$
• D. $-\frac{4}{3}$

Answer 1

The correct option is C $\frac{4}{3}$

Let $I=\underset{0}{\overset{\pi }{\int }}|\cos x{|}^{3}dx=2\underset{0}{\overset{\frac{\pi }{2}}{\int }}|\cos x{|}^{3}dx$
$[\because |\cos (\pi -x)|=|-\cos x|=|\cos x|]$

$\therefore I=2\underset{0}{\overset{\frac{\pi }{2}}{\int }}|\cos x{|}^{3}dx=2\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\cos }^{3}xdx$
$\Rightarrow I=2\underset{0}{\overset{\frac{\pi }{2}}{\int }}(1-{\sin }^{2}x)\cos xdx=2\underset{0}{\overset{\frac{\pi }{2}}{\int }}(\cos x-\cos x{\sin }^{2}x)dx=2{[\sin x-\frac{{\sin }^{3}x}{3}]}_0^{\frac{\pi }{2}}=2(1-\frac{1}{3})=\frac{4}{3}$
Hence, $\underset{0}{\overset{\pi }{\int }}|\cos x{|}^{3}dx=\frac{4}{3}$