$\begin{align}  & \int{x{{e}^{-x}}dx} \\ & \text{Using integration by part} \\ & \int{udv}=uv-\int{vdu} \\ & \text{Let }u=x,\text{ }du=dx \\ & dv={{e}^{-x}}dx,\text{ }v=-{{e}^{-x}} \\ & \int{x{{e}^{-x}}dx}=x(-{{e}^{-x}})-\int{(-{{e}^{-x}})dx} \\ & \int{x{{e}^{-x}}dx}=-x{{e}^{-x}}+\int{{{e}^{x}}dx}=-x{{e}^{-x}}-{{e}^{-x}}+C \\\end{align}$