Question

How do you integrate `int x^3 cot x dx` using integration by parts?

Answer 1

You can't.

Explanation:

The antiderivative of `x^3cotx` involves the Polylogarithm Function evaluated at imaginary values.

`I = intx^3cotx dx`

Let `u = x^3` so `du = 3x^2 dx` and

let `dv = cotx dx` so `v = intcotx dx = ln abssinx`.

`I = x^3lnabssinx - 3int x^2 ln abs sinx dx`

Let `u = x^2` so `du = 2x dx`

let `dv = ln abs sinx dx` so `v = 1/2i(x^2+Li_2(e^(2ix)))-xln(1-e^(2ix))+xlnabs sinx`

Where `Li_2(x) = sum_(k=1)^oo x^k/k^2` `" "` (known as the polylogarithm or Jonquiere's function.)

At this point I'll let you finish yourself. (Because I'm out of my depth.)