Answer
$$\frac{1}{3}\left( {x - 3} \right)\sqrt {2x + 3} + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{x}{{\sqrt {2x + 3} }}} dx \cr
& \int {\frac{x}{{\sqrt {2x + 3} }}} dx \cr
& {\text{Integrate by tables using the formula }} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\int {\frac{x}{{\sqrt {ax + b} }}} = \frac{2}{{3{a^2}}}\left( {ax - 2b} \right)\sqrt {ax + b} + C \cr
& ,{\text{then}} \cr
& \int {\frac{x}{{\sqrt {2x + 3} }}} dx = \frac{2}{{3{{\left( 2 \right)}^2}}}\left( {2x - 2\left( 3 \right)} \right)\sqrt {2x + 3} + C \cr
& {\text{Simplifying}} \cr
& \int {\frac{x}{{\sqrt {2x + 3} }}} dx = \frac{2}{{12}}\left( {2x - 6} \right)\sqrt {2x + 3} + C \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{4}{{12}}\left( {x - 3} \right)\sqrt {2x + 3} + C \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{3}\left( {x - 3} \right)\sqrt {2x + 3} + C \cr} $$
