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

Question

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

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Answer

Let $\mathrm{I}=\int_0^{\pi / 2} \cos ^3 x \cdot d x$
$
\begin{aligned}
& =\int_0^{\pi / 2} \frac{1}{4}[\cos 3 x+3 \cos x] \cdot d x \\
& =\frac{1}{4}\left[\sin 3 x \cdot \frac{1}{3}+3 \sin x\right]_0^{\pi / 2} \\
& =\frac{1}{4}\left[\left(\frac{1}{3} \sin 3 \frac{\pi}{2}+3 \sin \frac{\pi}{2}\right)-\right.
\end{aligned}
$
$
\left.\left(\frac{1}{3} \sin 3(0)+3 \sin (0)\right)\right]
$
$=\frac{1}{4}\left[\frac{1}{3} \sin \frac{3 \pi}{2}+3 \sin \frac{\pi}{2}-\right.$
$
\left.\frac{1}{3} \sin 0+3 \sin 0\right]
$
$=\frac{1}{4}\left[\frac{1}{3}(-1)+3(1)-0\right]$
$=\frac{1}{4}\left[-\frac{1}{3}+3\right]=\frac{1}{4}\left[\frac{8}{3}\right]=\frac{2}{3}$
$\therefore \quad \int_0^{\pi / 2} \cos ^3 x \cdot d x=\frac{2}{3}$

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