Chapter 8 - Techniques of Integration - 8.2 Trigonometric Integrals - Exercises - Page 403: 33

$$\frac{1}{2}\cot (3-2 x)+c$$

Given $$\int \csc ^{2}(3-2 x) d x$$ Let $$ u=3-2x\ \ \Rightarrow \ \ du=-2dx$$ Then \begin{align*} \int \csc ^{2}(3-2 x) d x&=\frac{1}{2} \int\left(-\csc ^{2} u\right) d u\\ &=\frac{1}{2}\cot u+c\\ &=\frac{1}{2}\cot (3-2 x)+c \end{align*}