Evaluate the following integrals: ^ 2 _0 ^ 2 x dx - Vidyadip

Question

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

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Answer

$\int\limits^\frac{\pi}{2}_0\sin^2\text{x}\text{dx}$
$= \int\limits^\frac{\pi}{2}_0\frac{1-\cos2\text{x}}{2}\text{dx}$
$= \frac{1}{2}\int\limits^\frac{\pi}{2}_0(1-\cos2\text{x})\text{dx}$
$=\frac{1}{2}\Big[\text{x}-\frac{\sin2\text{x}}{2}\Big]^\frac{\pi}{2}_0$
$=\frac{1}{2}(\frac{\pi}{2}-0)$
$=\frac{\pi}{4}$

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