Example 9. Find the integral $\int \frac{d x}{\sin ^{2} 3 x}$.
Solution.

$$
\int \frac{d x}{\sin ^{2} 3 x}=\int \frac{\frac{1}{3} d(3 x)}{\sin ^{2} 3 x}=\frac{1}{3} \int \frac{d(3 x)}{\sin ^{2} 3 x}=-\frac{1}{3} \operatorname{ctg}^{2} 3 x+C
$$