The value of

Question:

The value of $\int_{0}^{\pi}|\cos x|^{3} d x$

  1. $2 / 3$

  2. 0

  3. $-4 / 3$

  4. $4 / 3$


Correct Option: , 4

Solution:

$\int_{0}^{\pi}|\cos x|^{3} d x=\int_{0}^{\pi / 2} \cos ^{3} x d x-\int_{\pi / 2}^{\pi} \cos ^{3} x d x$

$=\int_{0}^{\pi / 2}\left(\frac{\cos 3 x+3 \cos x}{4}\right) d x-\int_{\pi / 2}^{\pi}\left(\frac{\cos 3 x+3 \cos x}{4}\right) d x$

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

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

$=\frac{4}{3}$

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