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

Question

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

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Answer

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

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