Question

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

• A. 2/3
• B. 0
• C. -4/3
• D. 4/3

Answer 1

Answer: (D)

${\int }_0^{\pi }{\left|\cos x\right|}^{3}dx={\int }_0^{\pi /2}{\cos }^{3}xdx-{\int }_{\pi /2}^{\pi }{\cos }^{3}xdx$
$={\int }_0^{\pi /2}\left(\frac{\cos 3x+3\cos x}{4}\right)dx-{\int }_{\pi /2}^{\pi }\left(\frac{\cos 3x+3\cos x}{4}\right)dx$
$=\frac{1}{4}\left[{\left(\frac{\sin 3x}{3}+3\sin x\right)}_0^{\pi /2}-{\left(\frac{\sin 3x}{3}+3\sin x\right)}_{\pi /2}^{\pi }\right]$
$=\frac{1}{4}\left[\left(\frac{-1}{3}+3\right)-\left(0+0\right)-\left\{\left(0+0\right)-\left(\frac{-1}{3}+3\right)\right\}\right]$
$=\frac{4}{3}$