Answer
$$4 - \frac{4}{{{e^2}}}$$
Work Step by Step
$$\eqalign{
& \int_0^2 {4x{e^{ - {x^2}/2}}} dx \cr
& {\text{substitute }}u = - {x^2}/2,{\text{ }}du = - xdx \cr
& {\text{express the limits in terms of }}u \cr
& x = 2{\text{ implies }}u = - {\left( 2 \right)^2}/2 = - 2 \cr
& x = 0{\text{ implies }}u = - {\left( 0 \right)^2}/2 = 0 \cr
& {\text{the entire integration is carried out as follows}} \cr
& \int_0^2 {4x{e^{ - {x^2}/2}}} dx = - 4\int_0^{ - 2} {{e^u}} du \cr
& {\text{find the antiderivative}} \cr
& = - 4\left. {{e^u}} \right|_0^{ - 2} \cr
& {\text{use the fundamental theorem}} \cr
& = - 4\left( {{e^{ - 2}} - {e^0}} \right) \cr
& {\text{simplify}} \cr
& = 4 - \frac{4}{{{e^2}}} \cr} $$
