mean of a continuous random variable and integration by parts

Question

$g(t)= \ln(t)$ for $1\leq t \leq e$ is a PDF for a continuous random variable $T$. find the mean of $T$ using the definition of the mean of a continuous random variable and then performing integration by parts.

I don't seem to understand how to do this?

Answer 1

$\int_1^e t\ln(t)dt$

This is a canonical integration by parts problem

Answer 2

As Prototank says, this is a canon IBP. Set $u = \ln t$ and $dv = t \ dt$. Can you go from here?