Explanation

(c)
Let $I = \int_{0}^{\pi/2}\frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}}dx$ …(i)
$\Rightarrow I = \int_{0}^{\pi/2}{\frac{2^{\cos x}}{2^{\cos x}2^{\sin x}}dx}$ ….(ii)
$\left\lbrack put\ x = \left( \frac{\pi}{2} - x \right) \right\rbrack$
On adding Eqs. (i) and (ii), we get
$2I = \int_{0}^{\pi/2}{1dx = \frac{\pi}{2}}$
$\Rightarrow \ I = \frac{\pi}{4}$