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.
$\displaystyle (1+4cot(x))/(4-cot(x))$
Thanks for any help you can offer!
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.
$\displaystyle (1+4cot(x))/(4-cot(x))$
Thanks for any help you can offer!
$\displaystyle \frac{1 + 4\cot(x)}{4 - \cot(x) }$
$\displaystyle = \frac{ \tan(x) \cdot (1 + 4\cot(x) )}{ \tan(x) \cdot ( 4 - \cot(x))} $
$\displaystyle = \frac{ \tan(x) + 4 }{ 4\tan(x) - 1 } $
$\displaystyle = -\frac{ \tan(x) + 4}{ 1 - 4\tan(x) }$
then use the formula
$\displaystyle \tan(A + B) = \frac{ \tan(A) + \tan(B) }{ 1 - \tan(A)\tan(B) } $
By substituting $\displaystyle A = x , B = \tan^{-1}(4) $
We obtain $\displaystyle \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 :
$\displaystyle \ln{ \left [ \cos( x + \tan^{-1}(4) ) \right ]} + C $