Need help with Trig Integral

**peachfuzz** · February 22nd 2010, 03:41 PM

Hello,

I've been trying to work the following problem, but I am getting lost from the start. I'm pretty sure if I could get it started I wouldn't have a problem with it.

$(1+4cot(x))/(4-cot(x))$

Thanks for any help you can offer!

**simplependulum** · February 22nd 2010, 11:06 PM

$\frac{1 + 4\cot(x)}{4 - \cot(x) }$

$= \frac{ \tan(x) \cdot (1 + 4\cot(x) )}{ \tan(x) \cdot ( 4 - \cot(x))}$

$= \frac{ \tan(x) + 4 }{ 4\tan(x) - 1 }$

$= -\frac{ \tan(x) + 4}{ 1 - 4\tan(x) }$

then use the formula

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

By substituting $A = x , B = \tan^{-1}(4)$

We obtain $\frac{1 + 4\cot(x)}{4 - \cot(x) } = - \tan( x + \tan^{-1}(4) )$

Wow , it is actually a tagent function so the integral is log. function :

$\ln{ \left [ \cos( x + \tan^{-1}(4) ) \right ]} + C$

**peachfuzz** · February 23rd 2010, 02:39 AM

Thank you!

Multiplying by 1 is something I would've never seen in my life. I always seem to forget that as an option.

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