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

1 Answer
Nov 28, 2016

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.)