Explanation

(c)
Given, $I = \int_{- \pi}^{\pi}\frac{e^{\sin}x}{e^{\sin{x\ }} + e^{- sinx}{\ \ }}dx$ …(i)
$\Rightarrow I = \int_{- \pi}^{\pi}\frac{e^{sin(2\pi - x)}}{e^{sin(2\pi - x)} + e^{- sin(2\pi - x)}}dx$
$\Rightarrow I = \int_{- \pi}^{\pi}{\frac{e^{- \sin x}}{e^{- \sin x} + e^{\sin x}}dx}$ …(ii)
On adding Eqs.(i) and (ii), we get
$2I = \int_{- \pi}^{\pi}{\frac{e^{\sin x} + e^{- \sin x}}{e^{sinx} + e^{- \sin x}}dx}$
$= \int_{- \pi}^{\pi}{1dx = \lbrack x\rbrack_{- \pi}^{\pi}}$
$I = \pi$