Evaluate the following integrals: ^ _0x ^2x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{\pi}_0\text{x}\cos^2\text{x dx}$

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Answer

Let $\text{I}=\int\limits^{\pi}_0\text{x}\cos^2\text{x dx}\ ...(\text{i})$
$=\int\limits^{\pi}_0(\pi-\text{x})\cos^2(\pi-\text{x})\text{dx}$
$=\int\limits^{\pi}_0(\pi-\text{x})\cos^2\text{x}\text{ dx}\ ...(\text{ii})$
Adding (i) and (ii) we get
$2\text{I}=\int\limits^{\pi}_0(\text{x}+\pi-\text{x})\cos^2\text{x}\text{ dx}$
$=\int\limits^{\pi}_0\pi\cos^2\text{x}\text{ dx}$
$=\pi\int\limits^{\pi}_0\frac{1+\cos2\text{x}}{2}\text{ dx}$
$=\frac{\pi}{2}\int\limits^{\pi}_0\big(1+\cos2\text{x}\big)\text{dx}$
$=\frac{\pi}{2}\Big[\text{x}+\frac{\sin2\text{x}}{2}\Big]^{\pi}_0$
$=\frac{\pi}{2}(\pi-0)$
Hence, $\text{I}=\frac{\pi^2}{4}$

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