Use integration by parts to evaluate the following integrals.

Question

∫x²lnxdx

For this one, do I set u=x^2 and dv=lnxdx? And so then would I use integration by parts to find v? Or would I set u=lnx and dv=x^2dx?

Answer 1

Let u = lnx, then dv = x²dx.

So, du = (1/x)dx and v = ∫dv = (1/3)x³

∫x²lnxdx = uv - ∫vdu = (1/3)x³lnx - (1/3)∫x²dx = (1/3)x³lnx - (1/9)x³ + C