Answer
$$\frac{{2{{\left( {4 - x} \right)}^{5/2}}}}{5} - \frac{{8{{\left( {4 - x} \right)}^{3/2}}}}{3} + C$$
Work Step by Step
$$\eqalign{
& \int {x\sqrt {4 - x} } dx \cr
& {\text{substitute }}u = 4 - x,{\text{ }}du = - dx \cr
& \int {x\sqrt {4 - x} } dx = \int {\left( {4 - u} \right)\sqrt u } \left( { - du} \right) \cr
& = \int {\left( {u - 4} \right)\sqrt u } du \cr
& = \int {\left( {{u^{3/2}} - 4{u^{1/2}}} \right)} du \cr
& {\text{find antiderivative }} \cr
& = \frac{{{u^{5/2}}}}{{5/2}} - 4\left( {\frac{{{u^{3/2}}}}{{3/2}}} \right) + C \cr
& = \frac{{2{u^{5/2}}}}{5} - \frac{{8{u^{3/2}}}}{3} + C \cr
& {\text{write in terms of }}x,{\text{ replace }}u = 4 - x \cr
& = \frac{{2{{\left( {4 - x} \right)}^{5/2}}}}{5} - \frac{{8{{\left( {4 - x} \right)}^{3/2}}}}{3} + C \cr} $$
