$\int\limits_0^1 {{e^{2x}}xdx} = \int\limits_0^1 {x.d\left( {\frac{1}{2}{e^{2x}}} \right)} =
x.\frac{1}{2}{e^{2x}}\left| \begin{array}{l}1\\0
\end{array} \right. - \frac{1}{2}\int\limits_0^1 {{e^{2x}}dx} $
$=\frac{e^2}{2}-\frac{1}{4}e^{2x}|^1_0=\frac{e^2+1}{4}$