Explanation

(c)
$\int_{}^{}e^{x}\left( \frac{1 - \sin x}{1 - \cos x} \right)dx = \int_{}^{}e^{x}\left( \frac{1 - {2\ sin}\frac{x}{2}\cos\frac{x}{2}}{2\sin^{2}\frac{x}{2}} \right)dx$
$= \frac{1}{2}\int_{}^{}e^{x}\left( cosec^{2}\ \frac{x}{2} \right)dx - \int_{}^{}e^{x}\cot\frac{x}{2}\ dx$
$= \frac{1}{2}\left\lbrack - e^{x}\cot\frac{x}{2}.2 + \int_{}^{}e^{x}\cot\frac{x}{2}.2\ dx \right\rbrack - \int_{}^{}e^{x}\cot\frac{x}{2}\ dx + c$
$= - e^{x}\cot\frac{x}{2} + c$