the integral of Ln x = x Lnx - x, but...?

Question

what is the integral of (Ln x)^2?

Answer 1

∫ln² x dx

Let u=ln² x, du = 2 ln x/x dx, dv=dx, v=x:

x ln² x - ∫2 ln x dx

Let u=2 ln x, du = 2/x dx, dv = dx, v=x:

x ln² x - 2 x ln x + ∫2 dx
x ln² x - 2 x ln x + 2x + C

And we are done.

Answer 2

Int = x(ln x)^2 -2Int (ln x dx) = x (ln x) ^2- 2(x lnx -x)

Answer 3

Integrate by parts.

dx = u'
x = u
(ln(x))^2 = v
2(ln(x))(1/x) dx = v'

You will get, when you are done

x((ln(x) - 2)ln(x) + 2)

HTH

Charles