Answer
$\frac{1}{4} (\tan 2\theta)^2+c$
Work Step by Step
Since $u=\tan 2\theta $, then $du=2\sec^2 2\theta d\theta $. Now, we have
$$
\int \sec^2(2\theta) \tan(2\theta) d\theta=\frac{1}{2}\int udu=\frac{1}{4} u^2+c\\
=\frac{1}{4} (\tan 2\theta)^2+c.
$$
