Question

The value of ${\int }_0^{\pi }|\cos x{|}^{3}dx$.

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

Answer 1

Answer: (D)

$\frac{4}{3}$

${\int }_0^{\pi }|\cos x{|}^{3}dx={\int }_0^{\pi /2}{\cos }^{3}xdx-{\int }_{\pi /2}^{\pi }{\cos }^{3}xdx$

$={\int }_0^{\pi /2}\left(\frac{\cos 3x+3\cos x}{4}\right)dx-{\int }_{\pi /2}^{\pi }\left(\frac{\cos 3x+3\cos x}{4}\right)dx$

$=\frac{1}{4}[{(\frac{\sin 3x}{3}+3\sin x)}_0^{\pi /2}-{(\frac{\sin 3x}{3}+3\sin x)}_{\pi /2}^{\pi }]$

$=\frac{1}{4}[(\frac{-1}{3}+3)-(0+0)-\left\{\right(0+0)-(\frac{-1}{3}+3)\}]$

$=\frac{4}{3}$.