u-substitution problem help

I need help with a u-substitution problem. It is has already been worked out for me but I don't understand how to get the final answer.

Here is the problem:

∫ln(x)/x

here is the worked out answer:

u=ln(x) ---> ∫udu=(u^2)/2 + C

=(ln(x))^2)/2 + C

I don't understand why ln(x) changes to u^2 when you substitute u into the integral, and I also don't understand where the 2 on the bottom comes from. I thought the substitution should be u/x if you substitute u for ln(x) into the integral. Maybe a step was skipped in the solution and that's why I don't understand? Anyway, if anyone could explain to me why it becomes u^2/2 + C instead of u/x, I would greatly appreciate it. Thanks in advance to anyone who can help.