Chapter 8 - Integration Techniques, L'Hopital's Rule, and Improper Integrals - 8.2 Exercises - Page 522: 64

$$\eqalign{ & \left( {\text{a}} \right){\text{ }} - \frac{2}{{15}}{\left( {4 - x} \right)^{3/2}}\left( {3x + 8} \right) + C \cr & \left( {\text{b}} \right){\text{ }} - \frac{2}{{15}}{\left( {4 - x} \right)^{3/2}}\left( {3x + 8} \right) + C \cr} $$

$$\eqalign{ & \int {x\sqrt {4 - x} } dx \cr & \left( {\text{a}} \right){\text{ By parts}} \cr & {\text{Let }}u = x,{\text{ }}du = dx \cr & {\text{ }}dv = \sqrt {4 - x} ,{\text{ }}v = - \frac{2}{3}{\left( {4 - x} \right)^{3/2}} \cr & \int {udv} = uv - \int {vdu} \cr & = - \frac{2}{3}x{\left( {4 - x} \right)^{3/2}} + \int {\frac{2}{3}{{\left( {4 - x} \right)}^{3/2}}} dx \cr & = - \frac{2}{3}x{\left( {4 - x} \right)^{3/2}} + \frac{2}{3}\left( {\frac{{{{\left( {4 - x} \right)}^{5/2}}}}{{5/2}}} \right) + C \cr & = - \frac{2}{3}x{\left( {4 - x} \right)^{3/2}} + \frac{4}{{15}}{\left( {4 - x} \right)^{5/2}} + C \cr & {\text{Factoring}} \cr & = - {\left( {4 - x} \right)^{3/2}}\left[ {\frac{2}{3}x + \frac{4}{{15}}\left( {4 - x} \right)} \right] + C \cr & = - {\left( {4 - x} \right)^{3/2}}\left( {\frac{2}{3}x + \frac{{16}}{{15}} - \frac{4}{{15}}x} \right) + C \cr & = - {\left( {4 - x} \right)^{3/2}}\left( {\frac{2}{5}x + \frac{{16}}{{15}}} \right) + C \cr & = - \frac{2}{{15}}{\left( {4 - x} \right)^{3/2}}\left( {3x + 8} \right) + C \cr & \cr & \left( {\text{b}} \right) \cr & {\text{Let }}u = 4 - x,{\text{ }}x = 4 - u,{\text{ }}dx = - du \cr & {\text{Substituting}} \cr & \int {x\sqrt {4 - x} } dx = \int {\left( {4 - u} \right)\sqrt u } \left( { - du} \right) \cr & = \int {\left( {u - 4} \right)\sqrt u } du \cr & = \int {\left( {{u^{3/2}} - 4{u^{1/2}}} \right)} du \cr & = \frac{{{u^{5/2}}}}{{5/2}} - \frac{{4{u^{3/2}}}}{{3/2}} + C \cr & = \frac{{2{u^{5/2}}}}{5} - \frac{{8{u^{3/2}}}}{3} + C \cr & {\text{Factoring}} \cr & = \frac{2}{{15}}{u^{3/2}}\left( {3u - 20} \right) + C \cr & {\text{Write in terms of }}x \cr & = \frac{2}{{15}}{\left( {4 - x} \right)^{3/2}}\left( {3\left( {4 - x} \right) - 20} \right) + C \cr & = \frac{2}{{15}}{\left( {4 - x} \right)^{3/2}}\left( {12 - 3x - 20} \right) + C \cr & = \frac{2}{{15}}{\left( {4 - x} \right)^{3/2}}\left( { - 3x - 8} \right) + C \cr & = - \frac{2}{{15}}{\left( {4 - x} \right)^{3/2}}\left( {3x + 8} \right) + C \cr} $$