TITLE: How to integrate $\int_1^e \ln{x} \, dx$
QUESTION [7 upvotes]: This seems really tricky to me. I can't figure out how to integrate $\ln x$.

REPLY [2 votes]: $$
\int\ln x\,dx=\int u\,dx = xu-\int x\,du = x\ln x - \int x\left(\frac1x\,dx\right).
$$
Now here's the hard part: $x\cdot\dfrac1x$ simplifies to $1$.  At least, I've seen lots of students get stuck on that part.  Some of them want to antidifferentiate $x$ and also $\dfrac1x$.
So you've got
$$
x\ln x-\int 1\,dx.
$$
You can probably do the rest.