# Math Help - Another Integration problem

1. ## Another Integration problem

1/[(cot(x/2)cot(x/3)cot(x/6)] .Find the integral with respect to x

2. ## Re: Another Integration problem

$\hspace{-16}\bf{\int \frac{1}{\cot(x).\cot\left(\frac{x}{2}\right).\cot\left(\frac{x}{3}\right)}dx = \int \tan(x).\tan\left(\frac{x}{2}\right).\tan\left(\frac{x}{3}\right)dx}\\\\\\ Put \bf{x=6t\Leftrightarrow dx=6dt}\\\\\\ \bf{6.\int \tan(t).\tan(2t).\tan(3t)dt}\\\\\\ Now \bf{3t=(2t+t)\Leftrightarrow \tan(3t)=\tan(2t+t)}\\\\\\ We Get \bf{\tan(t).\tan(2t).\tan(3t)=\tan(3t)-\tan(2t)-\tan(t)}\\\\\\ So Integral Convert into \bf{\int \left(\tan(3t)-\tan(2t)-\tan(t)\right)dt}\\\\\\ \bf{=\frac{1}{3}\ln\mid \sec (3t)\mid-\frac{1}{2}\ln\mid \sec (2t)\mid-\ln \mid \sec (t)\mid+C}\\\\\\ So \bf{ \int \tan(x).\tan\left(\frac{x}{2}\right).\tan\left(\frac{x}{3}\right)dx = \frac{1}{3}\ln\mid \sec \frac{x}{2}\mid-\frac{1}{2}\ln\mid \sec \frac{x}{3}\mid-\ln \mid \sec \frac{x}{6}\mid+C}$

3. ## Re: Another Integration problem

Thanks a lot I'd been stuck upon that for 20min