Evaluate the following integrals:

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 $\int \tan x d x=\log \sec x$

$y=\frac{x}{2} \tan 2 x-\frac{\log (\sec 2 x)}{4}+c$