Explanation

(c)
Given, $\cos y\frac{dy}{dx} = e^{{x + \sin}y} + x^{2}e^{\sin y}$
$\Rightarrow$ $\cos y\ \frac{dy}{dx} = e^{\sin y}\left( e^{x} + x^{2} \right)dx$
$\Rightarrow$ $\int_{}^{}{\frac{\cos y}{e^{\sin y}}\ dy = \int_{}^{}{(e^{x} + x^{2})}}dx$
Put $\sin y = t$ in LHS$\Rightarrow \cos y\ dy = dt$
$\therefore$ $\int_{}^{}\frac{dt}{e^{t}} = \int_{}^{}{(e^{x} + x^{2})}dx$
$\Rightarrow$ $- e^{- t} = e^{x} + \frac{x^{3}}{3} - c$
$\Rightarrow$ $e^{x} + e^{\sin y} + \frac{x^{3}}{3} = c$