Como \(\frac{y}{\textrm{e}^{2y}}=y\textrm{e}^{-2y}\), calculamos

\[ \int_0^1 y\textrm{e}^{-2y}\,dy. \]

Usamos integração por partes com

\[ u=y, \qquad dv=\textrm{e}^{-2y}\,dy. \]

Então \(du=dy\) e \(v=-\frac{1}{2}\textrm{e}^{-2y}\). Logo,

\[ \int y\textrm{e}^{-2y}\,dy =-\frac{y}{2}\textrm{e}^{-2y} +\frac{1}{2}\int \textrm{e}^{-2y}\,dy. \]

Portanto,

\[ \int y\textrm{e}^{-2y}\,dy =-\frac{y}{2}\textrm{e}^{-2y}-\frac{1}{4}\textrm{e}^{-2y}. \]

Avaliando de \(0\) até \(1\),

\[ \int_0^1 \frac{y}{\textrm{e}^{2y}}\,dy =\left[-\frac{y}{2}\textrm{e}^{-2y}-\frac{1}{4}\textrm{e}^{-2y}\right]_0^1. \]

\[ \colorbox{green!20}{ \int_0^1 \frac{y}{\textrm{e}^{2y}}\,dy =\frac{1}{4}-\frac{3}{4\textrm{e}^2} }. \]