Question

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

• A. 2/3
• B. 0
• C. -4/3
• D. 4/3

Answer 1

Answer: (D)

$\int^{\pi}_{0} \left|\cos x\right|^{3} dx = \int^{\pi/2}_{0} \cos^{3}x dx - \int^{\pi}_{\pi/2}\cos^{3} x dx $
$ = \int^{\pi/2}_{0} \left(\frac{\cos3x+3\cos x}{4}\right) dx - \int^{\pi}_{\pi/2}\left(\frac{\cos 3x+3\cos x}{4}\right) dx$
$ = \frac{1}{4} \left[\left(\frac{\sin3x}{3} + 3\sin x\right)^{\pi/2}_{0} -\left(\frac{\sin3x}{3} + 3\sin x\right)^{\pi}_{\pi/2}\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} $