Explanation

(c)
$\int_{}^{}\frac{e^{x}(1 + \sin{x)}}{(1 + \cos{x)}}dx$
$= \int_{}^{}{e^{x}\ \left\lbrack \frac{1}{2}\sec^{2}{\frac{x}{2} + \tan{\frac{x}{2}\ }} \right\rbrack dx}$
$= \frac{1}{2}\int_{}^{}{e^{x}{\sec^{2}\frac{x}{2}}{dx + \int_{}^{}{e^{x}\tan{\frac{x}{2}dx}}}}$
$= e^{x}\tan{\frac{x}{2} - \int_{}^{}{e^{x}\tan{\frac{x}{2}dx + \int_{}^{}{e^{x}\tan{\frac{x}{2}dx}}}}}$
$= e^{x}\tan{\frac{x}{2} + c}$
But $I = e^{x}f(x) + c$ (given)
$\therefore f(x) = \tan\frac{x}{2}$