Answer
$$ - \frac{{2\left( {8 - 5x} \right)}}{{75}}\sqrt {4 + 5x} + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{x}{{\sqrt {4 + 5x} }}} dx \cr
& {\text{From the table of integrals in the back of the book}} \cr
& \int {\frac{u}{{\sqrt {a + bu} }}} du = - \frac{{2\left( {2a - bu} \right)}}{{3{b^2}}}\sqrt {a + bu} + C,{\text{ }}\left( {\bf{1}} \right) \cr
& {\text{Comparing}} \cr
& \int {\frac{x}{{\sqrt {4 + 5x} }}} dx = \int {\frac{u}{{\sqrt {a + bu} }}} du \to a = 4,{\text{ }}b = 5,{\text{ }}u = x \cr
& {\text{Then substituting the values of }}a,b{\text{ and }}x{\text{ into }}\left( {\bf{1}} \right) \cr
& \int {\frac{x}{{\sqrt {4 + 5x} }}} dx = - \frac{{2\left( {2\left( 4 \right) - 5x} \right)}}{{3{{\left( 5 \right)}^2}}}\sqrt {4 + 5x} + C \cr
& {\text{Simplify}} \cr
& \int {\frac{x}{{\sqrt {4 + 5x} }}} dx = - \frac{{2\left( {8 - 5x} \right)}}{{75}}\sqrt {4 + 5x} + C \cr} $$
