Answer
$$ - x{e^{ - x}} - {e^{ - x}} + C$$
Work Step by Step
$$\eqalign{
& \int {x{e^{ - x}}} dx \cr
& {\text{substitute }}u = x,{\text{ }}du = dx \cr
& dv = {e^{ - x}}dx,{\text{ }}v = - {e^{ - x}} \cr
& {\text{ integration by parts}} \cr
& \int {udv} = uv - \int {vdu} \cr
& {\text{we have}} \cr
& \int {x{e^{ - x}}} dx = - x{e^{ - x}} - \int {\left( { - {e^{ - x}}} \right)dx} \cr
& \int {x{e^{ - x}}} dx = - x{e^{ - x}} + \int {{e^{ - x}}dx} \cr
& {\text{find antiderivative}} \cr
& \int {x{e^{ - x}}} dx = - x{e^{ - x}} + \left( { - {e^{ - x}}} \right) + C \cr
& \int {x{e^{ - x}}} dx = - x{e^{ - x}} - {e^{ - x}} + C \cr} $$
