Explanation

(c)
Let $I = \int_{\pi/4}^{\pi/2}{e^{x}(\log{\sin x} + \cot{x)dx}}$
$\Rightarrow \ I = \int_{\pi/4}^{\pi/2}{e^{x}\log{\sin{xdx + \int_{\pi/4}^{\pi/2}e^{x}\cot{xdx}}}}$
$= \int_{\pi/4}^{\pi/2}e^{x}\log{\sin{xdx + \lbrack e^{x}\log{\sin{x\left. \ \ \right\rbrack_{\pi/4}^{\pi/2}\ \ }}}}$
$- \int_{\pi/4}^{\pi/2}e^{x}log\ sin\ x\ dx$
$= e^{\pi/2}\log{\sin{\frac{\pi}{2} - e^{\pi/4}\log{\sin\frac{\pi}{4}}}}$
$= - e^{\pi/4}log\ \left( \frac{1}{\sqrt{2}} \right) = \frac{1}{2}e^{\pi/4}\log 2$