Tomamos

\[ u=x, \qquad dv=\textrm{e}^{-x}\,dx. \]

Assim \(du=dx\) e \(v=-\textrm{e}^{-x}\). Pela fórmula de integração por partes,

\[ \int x\textrm{e}^{-x}\,dx = -x\textrm{e}^{-x} - \int\left(-\textrm{e}^{-x}\right)\,dx. \]

Portanto,

\[ \colorbox{green!20}{\int x\textrm{e}^{-x}\,dx = -(x+1)\textrm{e}^{-x}+C}. \]