The value of ${\displaystyle \underset{e^{-1}}{\overset{e^2}{\int }}\left|\frac{\log x}{x}\right|}dx$ is:

${\displaystyle \underset{e^{-1}}{\overset{e^2}{\int }}\left|\frac{\log x}{x}\right|}dx$ = $-{\displaystyle \underset{e^{-1}}{\overset{1}{\int }}\frac{\log x}{x}}dx+{\displaystyle \underset{1}{\overset{e^2}{\int }}\frac{\log x}{x}}dx$

(since $\log x<0$ for $x\in \left[e^{-1},1\right]$ and $\log x>0$ for $x\in \left(1,e^2\right)$

                    =${\displaystyle \underset{-1}{\overset{0}{\int }}tdt+}-{\displaystyle \underset{1}{\overset{2}{\int }}tdt=-\frac{t^2}{2}{\left|{}_{-1}^0+\frac{t^2}{2}\right|}_0^2}=\frac{1}{2}+2=\frac{5}{2}$