# Integration

• Mar 30th 2013, 04:23 AM
smokesalot
Integration
$\int tan(x)tan(2x)tan(3x)dx$
• Mar 30th 2013, 04:56 AM
MINOANMAN
Re: Integration
• Mar 30th 2013, 06:02 AM
smokesalot
Re: Integration
Actually, this is a prove question, I really need steps

thanks
• Mar 30th 2013, 07:07 AM
MINOANMAN
Re: Integration
I need to leave ....
But try tan(2x)=2tanx/(1-(tanx)^2) , tan(3x)=(3tanx-(tanx)^3)/(1-3(tanx)^2)
• Mar 30th 2013, 12:54 PM
MINOANMAN
Re: Integration
Smoke..
Just mow I came back home and I started to solve your integral...
I have discovered a very interesting result
believe it or not (tanx)(tan2x)(tan3x) = tan(3x)-tan(2x)-tanx
indeed :
tan(3x) = (tanx+tan(2x))/(1-tanxtan(2x))
this implies : tan(3x)(1-tanxtan(2x))=tanx+tan(2x) and after all simplifications we obtain : tan(3x)-tan(2x)-tan(x) = tanxtan(2x)tan(3x)

therefore your integral now is simple to calculate:
since integral(tanx)dx = -ln(cosx)
integral(tan(2x))dx = (-1/2)ln(cos(2x))
and Integral (tan(3x))dx = (-1/3)ln(cos(3x))
you can verify all these and check the calculations..

MINOAS
• Apr 1st 2013, 01:40 AM
smokesalot
Re: Integration
Thank you!
• Apr 2nd 2013, 11:39 AM
MINOANMAN
Re: Integration
ok