Question:
Evaluate $\int x \sec ^{2} 2 x d x$
Solution:
Use method of integration by parts
$y=x \int \sec ^{2} 2 x d x-\int \frac{d}{d x} x\left(\int \sec ^{2} 2 x d x\right) d x$
$y=x \frac{\tan 2 x}{2}-\int \frac{\tan 2 x}{2} d x$
Use formula $\int \tan x d x=\log \sec x$
$y=\frac{x}{2} \tan 2 x-\frac{\log (\sec 2 x)}{4}+c$