Evaluate the definite integral _0^ 2 ^2 x dx - Vidyadip

Question

Evaluate the definite integral $\int\limits_0^{\frac{\pi }{2}} {{{\cos }^2}x dx} $

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Answer

$\int\limits_0^{\frac{\pi }{2}} {{{\cos }^2}xdx} $ $= \int\limits_0^{\frac{\pi }{2}} {\frac{{1 + \cos 2x}}{2}dx} $
$= \frac{1}{2}\int\limits_0^{\frac{\pi }{2}} {\left( {1 + \cos 2x} \right)dx} $
$= \frac{1}{2}\left( {x + \frac{{\sin 2x}}{2}} \right)_0^{\frac{\pi }{2}}$
$= \frac{1}{2}\left[ {\frac{\pi }{2} + \frac{1}{2}\sin \pi - \left( {0 + \frac{1}{2}\sin {0^o}} \right)} \right]$
$= \frac{1}{2}\left[ {\frac{\pi }{2} + 0 - 0} \right]$
$ = \frac{\pi }{4}$

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