7.2 definite integral — ગણિત ધોરણ 12 સાયન્સ — Question

$I = \int_0^{\,\pi /2} {\frac{{{2^{\sin \left( {\frac{\pi }{2} - x} \right)}}}}{{{2^{\sin \left( {\frac{\pi }{2} - x} \right)}} + {2^{\cos \left( {\frac{\pi }{2} - x} \right)}}}}dx} $

$ = \int_0^{\pi /2} {\frac{{{2^{\cos x}}}}{{{2^{\cos x}} + {2^{\sin x}}}}} \,dx$....$(ii)$

Adding equations $(i)$ and $(ii),$ we get

$2I = \int_0^{\pi /2} {\left( {\frac{{{2^{\sin x}} + {2^{\cos x}}}}{{{2^{\sin x}} + {2^{\cos x}}}}} \right)dx = \int_{\,0}^{\,\pi /2} {1\,dx} = [x]\,_0^{\pi /2} = \frac{\pi }{2}} $

Therefore, $I = \frac{\pi }{4}$.