Which expression should be set as u and what is the minimum number of applications of integration by parts that can be used to evaluate this integral?

Question

Consider the integral ∫x² ln(x) dx and the integration by parts method ∫udv=uv − ∫vdu + C.

Which expression should be set as u and what is the minimum number of applications of integration by parts that can be used to evaluate this integral?

u = ln(x) with one application

u = ln(x) with two applications

u = x² with one application

u = x² with two applications

Answer 1

When doing integration by parts, you choose u to be the function that, if you take enough derivatives, will eventually produce a constant. ln(x) will never do that (dln(x)/dx = 1/x, d(1/x)/dx = -1/x^2 ...)

Now how many derivatives do you need to take x^2 to get a constant value?