The integral \({\rm{\;}}\mathop \smallint \nolimits_1^{\rm{e}} \left\{ {{{\left( {\frac{{\rm{x}}}{{\rm{e}}}} \right)}^{2{\rm{x}}}} - {{\left( {\frac{{\rm{e}}}{{\rm{x}}}} \right)}^{\rm{x}}}} \right\}{\rm{lo}}{{\rm{g}}_{\rm{e}}}{\rm{xdx\;is}}\) equal to

• A. \(\frac{3}{2} - {\rm{e}} - \frac{1}{{2{{\rm{e}}^2}}}\)
• B. \( - \frac{1}{2} + \frac{1}{{\rm{e}}} - \frac{1}{{2{{\rm{e}}^2}}}\)
• C. \({\rm{\;}}\frac{1}{2} - {\rm{e}} - \frac{1}{{{{\rm{e}}^2}}}\)
• D. \(\frac{3}{2} - \frac{1}{{\rm{e}}} - \frac{1}{{2{{\rm{e}}^2}}}\)