In the section of integration by parts
Find the area of the region enclosed by y=lnx, the line x=e and the x-axis
I expect that this is the region enclosed by the function, the line $\displaystyle x = e$ and the $\displaystyle x$ intercept of the function (which is $\displaystyle x = 1$).
So you are evaluating
$\displaystyle \int_1^e{\ln{x}\,dx} = \int_1^e{1\cdot \ln{x}\,dx}$.
Now set $\displaystyle u = \ln{x}$ and $\displaystyle dv = 1$.