Answer
$$x{e^x} - {e^x} + C$$
Work Step by Step
$$\eqalign{
& \int {x{e^x}} dx \cr
& {\text{setting }}\,\,\,\,\,\,u = x{\text{ then }}du = dx\,\,\,\,\,\,\,\,\,\, \cr
& {\text{and}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,dv = {e^x}dx{\text{ then }}v = {e^x} \cr
& {\text{Substituting these values into the formula for integration by parts}} \cr
& \int u dv = uv - \int {vdu} \cr
& \int {x{e^x}} dx = \left( x \right)\left( {{e^x}} \right) - \int {\left( {{e^x}} \right)} dx \cr
& \int {x{e^x}} dx = x{e^x} - \int {{e^x}} dx \cr
& {\text{integrate}} \cr
& \int {x{e^x}} dx = x{e^x} - {e^x} + C \cr} $$
