Question

Solve:
$\int {\tan }^{5}2x\sec 2xdx$

Answer 1

$I=\int {\tan }^{5}2x\sec 2xdx$

$I=\int {\tan }^{4}2x\tan 2x\sec 2xdx$

Let $t=\tan 2x\Rightarrow dt=2\sec 2x\tan 2xdx\Rightarrow \sec 2x\tan 2xdx=\frac{dt}{2}$

$=\frac{1}{2}\int t^{4}dt$

$=\frac{1}{2}\frac{t^5}{5}+c$ where $c$ is the constant of integration.

$=\frac{1}{10}{\tan }^{5}2x+c$ where $t=\tan 2x$