Question

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

Answer 1

${\int }_{e^{-1}}^{e^2}\mid \frac{logx}{x}\mid dx$

$={\int }_{e^{-1}}^{e^2}\frac{logx}{x}dx$

put $\log x=t$

$\frac{1}{x}dx=dt$

$={\int }_{-1}^{2}tdt$

$={\left[\frac{t^2}{2}\right]}_{-1}^2$

$=\frac{1}{2}[4-1]=\frac{3}{2}$