Evaluate the following integrals: ^ ( )^2 3 _ 0 x ^2x^ 3 2 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{(\pi)^\frac{2}{3}}_{0}\sqrt{\text{x}}\cos^2\text{x}^{\frac{3}{2}}\text{ dx}$

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Answer

Let $\text{I}=\int^\limits{(\pi)^\frac{2}{3}}_{0}\sqrt{\text{x}}\cos^2\text{x}^{\frac{3}{2}}\text{ dx}$ Then,
Let $\text{x}^{\frac{3}{2}}=\text{t}$ Then, $\frac{3}{2}\sqrt{\text{x}}\text{ dx}=\text{dt}$
When, $\text{x}=0,\text{t}=0$ and $\text{x}=\big(\pi\big)^{\frac{2}{3}},\text{t}=\pi$
$\therefore\ \text{I}=\frac{2}{3}\int^\limits{\pi}_{0}\cos^2\text{t}\text{ dt}$
$\Rightarrow\text{I}=\frac{2}{3}\int^\limits{\pi}_{0}\frac{1+\cos2\text{x}}{2}\text{ dx}$
$\Rightarrow\text{I}=\frac{1}{3}\Big[\text{x}+\frac{\sin2\text{x}}{2}\Big]^{\pi}_0$
$\Rightarrow\text{I}=\frac{1}{3}\big(\pi+0\big)$
$\Rightarrow\text{I}=\frac{\pi}{3}$

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