Question

Hello,
I'm having trouble with this exercise. Can you help me?

Integral of (x* (csc x)^2)dx

I'm using the uv - integral v du formula. I tried with u= (csc x)^2 and used some trigonometric formulas, but the expression became too complicated, I couldn't continue working. Then I tried with u=x, but the same happened.

Thank you in advance.

You are on the right track using integration by parts. Let u = x and let dv = csc^2 x dx = (1/sin^2 x) dx
du = x
v = -cot x (You will have to prove that to yourself)
Integral u dv = uv - Integal v du
= - cot x - Integral (-cot x dx)
= - cot x + log sin x
You will have to prove that Integral of -cot x dx yourself also.

I have verified the steps and final answer with a table of integrals