Evaluate: _0^ 2 ^3 x d x - Vidyadip

Question

Evaluate: $\int_0^{\frac{\pi}{2}} \cos ^3 x d x$

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Answer

$\int_0^{\frac{\pi}{2}} \cos ^3 x d x=\int_0^{\frac{\pi}{2}}\left(\frac{\cos 3 x+3 \cos x}{4}\right) d x$
$=\frac{1}{4}\left[\int_0^{\frac{\pi}{2}} \cos x d x+3 \int_0^{\frac{\pi}{2}} \cos x d x\right]$
$=\frac{1}{4}\left[\left[\frac{\sin 3 x}{3}\right]_0^{\frac{\pi}{2}}+3[\sin x]_0^{\frac{\pi}{2}}\right]$
$=\frac{1}{4}\left[\frac{1}{3}\left(\sin \frac{3 \pi}{2}-\sin 0\right)+3\left(\sin \frac{\pi}{2}-\sin 0\right)\right]$
$=\frac{1}{4}\left[\frac{1}{3}(-1-0)+3(1-0)\right]$
$=\frac{1}{4}\left(\frac{-1}{3}+3\right)$
$=\frac{1}{4}\left(\frac{8}{3}\right)$
$=\frac{2}{3}$

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