Evaluate : \[\int_{\pi / 2}^{\pi} e^{x}\left(\frac{1-\sin x}{1-\cos x}\right) d x \]

\[\begin{aligned} I & =\int_{\pi / 2}^{\pi} e^{x}\left(\frac{1-\sin x}{1-\cos x}\right) d x \\ & =\int_{\pi / 2}^{\pi} e^{x}\left(\frac{1-2 \sin \dfrac{x}{2} \cos \dfrac{x}{2}}{2 \sin ^{2} \dfrac{x}{2}}\right) d x \\ & =\int_{\pi / 2}^{\pi} e^{x}\left(\frac{1}{2} \operatorname{cosec} e^{2} \frac{x}{2}-\cot \frac{x}{2}\right) d x \\ \therefore I & =-\left[e^{x} \cot \frac{x}{2}\right]_{\pi / 2}^{\pi} \\ & =-\left[e^{\pi} \cot \frac{\pi}{2}-e^{\frac{\pi}{2}} \cot \frac{\pi}{4}\right] \\ & =e^{\frac{\pi}{2}} \end{aligned} \]