Integrals — Maths STD 12 Science — Question

${\because \int\limits_{ - a}^a {f\left( x \right)dx = 2\int\limits_0^a {f\left( x \right)dx,} } }$ when f(x) is even function]
$\Rightarrow I = 2\int\limits_0^{\frac{\pi }{2}} {{{\sin }^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 = 2\int\limits_0^{\frac{\pi }{2}} {{{\cos }^2}xdx} $ …(ii)
Adding eq. (i) and (ii),
$2I = 2\int\limits_0^{\frac{\pi }{2}} {\left( {{{\sin }^2}x + {{\cos }^2}x} \right)dx} $
$= 2\int\limits_0^{\frac{\pi }{2}} {1dx} $
$= 2\left( x \right)_0^{\frac{\pi }{2}}$
$ = 2.\frac{\pi }{2} = \pi $
$\Rightarrow I = \frac{\pi }{2}$