How to integrate (Ln^2)(x)dx?

Answer 1

∫(ln x)^2 dx; use integration by parts, with u = ln x and dv = ln x dx. We know (or you can easily verify) that v = ∫ln x dx = x ln x - x. (To derive this result, use integration by parts where you integrate 1 dx and differentiate ln x.)
So we get
∫(ln x)^2 = ln x (x ln x - x) - ∫(x ln x - x)(1/x) dx
= x (ln x)^2 - x ln x - ∫(ln x - 1) dx
= x (ln x)^2 - x ln x - ((x ln x - x) - x) + c
= x (ln x)^2 - 2x ln x + 2x + c.