Question

Evaluate $\int tanx\tan 2x\tan 3xdx$.

Answer 1

$\int \tan x\tan 2x\tan 3xdx------\left(i\right)$

$\tan 3x=\tan (x+2x)$

$=\frac{\tan x+\tan 2x}{1-\tan x\tan 2x}$

$\left\{\tan \right(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}\}$

$\tan 3x-\tan x-\tan 2x=\tan x\tan 2x\tan 3x-------\left(ii\right)$

From equation $\left(i\right)$ and $\left(ii\right)$

$\int \tan 3x-\tan x-\tan 2xdx$

$\Rightarrow \frac{1}{3}\log \left|\sec \right(3x\left)\right|-\frac{1}{2}\log |\sec 2x|-\log |\sec x|+c$