Question:
The value of $\int_{0}^{\pi}|\cos x|^{3} \mathrm{~d} x$ is:
Correct Option: , 2
Solution:
$I=\int_{0}^{\pi}|\cos x|^{3} d x$
$=2 \int_{0}^{\pi / 2} \cos ^{3} x d x$
$=\frac{2}{4} \int_{0}^{\pi / 2}(3 \cos x+\cos 3 x) d x$
$\left[\because \cos 3 \theta=4 \cos ^{3} \theta-3 \cos \theta\right]$
$=\frac{1}{2}\left[3 \sin x+\frac{\sin 3 x}{3}\right]_{0}^{\pi / 2}$
$=\frac{1}{2}\left(3-\frac{1}{3}\right)=\frac{4}{3}$
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