Solving P(cotg x)^5: Can You Help Me Find the Primitive?

[joao_pimentel]

Hi people

Can you kindly tell me how to solve P(cotg x)^5 step by step?

I made some calculation but I couldn't achieve any result

Thank you :)

 

[LCKurtz]

Hi people

Can you kindly tell me how to solve P(cotg x)^5 step by step?

I made some calculation but I couldn't achieve any result

Thank you :)

These types of problems usually require repeated integration by parts or substitution or, better, use of a reduction formula. A reduction formula looks like this:

\int \cot^n(ax)\, dx = -\frac{\cot^{n-1}(ax)}{a(n-1)} - \int\cot^{n-2}(ax)\, dx

Once you show that you just use it repeatedly until n is small enough that you can finish the answer.

Try writing cotⁿ(ax) as cot^n-2(ax)(csc²(ax)-1) and use a u substitution to get the above identity. Then use it on your problem.

 

[joao_pimentel]

Thank you very much for your quick reply

I quite understood your answer and the reduction process, though I used another way to get the answer:

P(cotg(x)^5 ) = P (cos(x)^5/sen(x)^5) =
P( (cos(x).(1-sen(x)^2)^2) / sen(x)^5) =
P( cos(x)/sen(x)^5 - 2cos(x)sen(x)^2)/sen(x)^5 + cos(x)sen(x)^4/sen(x)^5)=
P(u'u^(-5) -2u'u^(-3)+u'u^(-1)) =
=-sen(x)^(-4)/4 +sen(x)^(-2) +log(|sen(x)

Thank you very much any way for your attention :)

kind regards

João