Explanation

(c)
Let $I = \int_{0}^{\pi/2}{\frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}}dx}$ …(i)
$\Rightarrow \ \ I = \int_{0}^{\pi/2}{\frac{2^{\sin{x(\pi/2 - x)}}}{2^{\sin{(\pi/2 - x)}} + 2^{\cos{x(\pi/2 - x)}}}dx}$
$\Rightarrow$ $I = \int_{0}^{\pi/2}{\frac{2^{\cos x}}{2^{\cos x} + 2^{\sin x}}dx}$ …(ii)
On adding Eqs.(i) and (ii), we get
$2I = \int_{0}^{\pi/2}{\frac{2^{\sin x} + 2^{\cos x}}{2^{\sin x} + 2^{\cos x}}dx = \frac{\pi}{2}}$
$\Rightarrow I = \frac{\pi}{4}$