Is x/(x-1)^2 integrable

yes, let u=x-1
or
rewrite x/(x-1)^2 = 1/(x-1) + 1/(x-1)^2 , it's very easy to integrate

Sure. The easy way is:

∫ (x/(x-1)^2) dx = ∫ (x-1 + 1) /(x-1)^2 dx
= ∫ (1/(x-1)) dx + ∫ (1/(x-1)^2) dx
= ln |x-1| - 1/(x-1) + C