What will be the integral of …

Let {tex}I = \int\limits_{\frac{{ - \pi }}{2}}^{\frac{\pi }{2}} {{{\sin }^2}xdx} {/tex}

{tex}= 2\int\limits_0^{\frac{\pi }{2}} {{{\sin }^2}xdx} {/tex} ...(i)

 {tex}{\because \int\limits_{ - a}^a {f\left( x \right)dx = 2\int\limits_0^a {f\left( x \right)dx,} } }{/tex} when f(x) is even function]

{tex}\Rightarrow I = 2\int\limits_0^{\frac{\pi }{2}} {{{\sin }^2}\left( {\frac{\pi }{2} - x} \right)dx} {/tex}

{tex}\left[ {\because \int\limits_0^a {f\left( x \right)dx = \int\limits_0^a {f\left( {a - x} \right)dx = } } } \right]{/tex}

{tex}\Rightarrow I = 2\int\limits_0^{\frac{\pi }{2}} {{{\cos }^2}xdx} {/tex} …(ii)

Adding eq. (i) and (ii),

{tex}2I = 2\int\limits_0^{\frac{\pi }{2}} {\left( {{{\sin }^2}x + {{\cos }^2}x} \right)dx} {/tex}

{tex}= 2\int\limits_0^{\frac{\pi }{2}} {1dx} {/tex}

{tex}= 2\left( x \right)_0^{\frac{\pi }{2}}{/tex}

{tex} = 2.\frac{\pi }{2} = \pi {/tex}

{tex}\Rightarrow I = \frac{\pi }{2}{/tex}