Chapter 8 - Techniques of Integration - 8.2 Trigonometric Integrals - Exercises - Page 403: 54

$$ -\ln |\sqrt{2}-1|$$

\begin{aligned} \int_{\pi / 4}^{\pi / 2} \frac{d x}{\sin x}&=\int_{\pi / 4}^{\pi / 2} \csc x d x \\ &= \ln |\csc x-\cot x|\bigg|_{\pi / 4} ^{\pi / 2} \\ &= \ln |1-0|-\ln |\sqrt{2}-1| \\&= -\ln |\sqrt{2}-1| \end{aligned}