Question

$lim\tan 2x.\tan 3x.\tan 5xdx$

Answer 1

Let $I=\int \tan 2x+\tan 3x+\tan 5xdx$

we know that,

$\tan 5x=\tan \left(3x+2x\right)$

$\Rightarrow \tan 5x=\frac{\tan 3x+\tan 2x}{1-\tan 3x\tan 2x}$

$\Rightarrow \tan 5x-\tan 5x\tan 3x\tan 2x=\tan 3x+\tan 2x$

$\Rightarrow \tan 5x-\tan 5x\tan 3x\tan 2x=\tan 3x+\tan 2x$

$\Rightarrow \int \tan 2x\tan 3x\tan 5xdx=\int \tan 5x-\tan 3x-\tan 2x$

$\Rightarrow I=\int \tan 5xdx-\int \tan 3xdx-\int \tan 2xdx$

$=\frac{\log \left|\sec 5x\right|}{5}-\frac{\log \left|\sec x\right|}{3}-\frac{\log \left|\sec 2x\right|}{2}$

Hence, we get,

$=\frac{1}{5}\log \left|\sec 5x\right|-\frac{1}{3}\log \left|sec3x\right|-\frac{1}{2}\log \left|\sec 2x\right|$