`int_1^e ((1-2\ ln\ x))/(x) dx`

Put `u = 1 - 2\ ln\ x`, then `du=(-2)/(x) dx`

`int_1^e ((1-2\ ln\ x))/(x) dx = -1/2 int_[x=1]^[x=e] u\ du`

`=-1/2[(u^2)/(2)]_[x=1]^[x=e]`

`=[(-u^2)/(4)]_[x=1]^[x=e]`

`=[(-(1-2\ ln\ x)^2)/(4)]_1^e`

`=[(-(1-2(1))^2)/(4)]-[(-(1-0)^2)/(4)]`

`=0`