understanding this improper integral

My teacher worked this problem one way and I think I have found an easier way. Also, I don't understand how he came to his conclusion.

Here is the problem: $\displaystyle \int_{0}^{e} ln {x}^2~dx$ This is how he worked the problem: $\displaystyle \lim_{b\to 0^+}\int_{b}^{e} 2~ln {x}~dx=2x~ln x -\int_{0}^{e} 2~dx=(2e-0)-2x (from~0~to~e)=2e-2e=0$. He also included $\displaystyle \lim_{x\to 0}\frac{2~lnx}{1/x}=\lim_{x\to 0^+}\frac{2/x}{-1/x^2}=\lim_{x\to 0^+}-2x=0$. This is what I did: $\displaystyle \lim_{b\to 0^+}2\int_{b}^{e} lnx~dx=\lim_{b\to 0^+}\frac{2}{x}~(from~b~to~e)=\frac{2}{e}-\frac{2}{b}=\frac{2}{e}$

Is my method correct? Any idea how he got what he did?

Thanks!