By using the properties of definite integrals, evaluate the integ… - Vidyadip

Question

By using the properties of definite integrals, evaluate the integral $\int\limits_0^{\frac{\pi }{2}} {{{\cos }^2}xdx} $

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Answer

Let $I = \int\limits_0^{\frac{\pi }{2}} {{{\cos }^2}xdx} $…(i)
$= \int\limits_0^{\frac{\pi }{2}} {{{\cos }^2}\left( {\frac{\pi }{2} - x} \right)dx} $
$\left[ {\because \int\limits_0^a {f\left( x \right)dx = \int\limits_0^a {f\left( {a - x} \right)dx = } } } \right]$
$ \Rightarrow I= \int\limits_0^{\frac{\pi }{2}} {{{\sin }^2}xdx} $ …(ii)
Adding equation (i) and (ii),
$2I = \int\limits_0^{\frac{\pi }{2}} {\left( {{{\cos }^2}x + {{\sin }^2}x} \right)dx} $
$= \int\limits_0^{\frac{\pi }{2}} {1dx} $
$\Rightarrow 2I = \left( x \right)_0^{\frac{\pi }{2}}$
$ \Rightarrow 2I = \frac{\pi }{2}$
$ \Rightarrow I = \frac{\pi }{4}$

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