Answer
$$\frac{3}{2}\left( {\sqrt {100 + 2{y^2}} - \sqrt 2 y} \right)$$
Work Step by Step
$$\eqalign{
& \int_0^5 {\frac{{6x}}{{\sqrt {4{x^2} + 2{y^2}} }}} dx \cr
& {\text{the notation }}dx{\text{ indicates integration with respect to }}x,{\text{ }} \cr
& {\text{so we treat }}x{\text{ as variable and }}y{\text{ as a constant}}{\text{. write the integrand as}} \cr
& = \int_0^5 {\frac{{6x}}{{{{\left( {4{x^2} + 2{y^2}} \right)}^{1/2}}}}} dx \cr
& = \int_0^5 {6x{{\left( {4{x^2} + 2{y^2}} \right)}^{ - 1/2}}} dx \cr
& = \frac{6}{8}\int_0^5 {{{\left( {4{x^2} + 2{y^2}} \right)}^{ - 1/2}}\left( {8x} \right)} dx \cr
& {\text{using the power rule }} \cr
& = \frac{3}{4}\left( {\frac{{{{\left( {4{x^2} + 2{y^2}} \right)}^{1/2}}}}{{1/2}}} \right)_0^5 \cr
& {\text{then}} \cr
& = \frac{3}{2}\left( {\sqrt {4{x^2} + 2{y^2}} } \right)_0^5 \cr
& {\text{evaluating the limits in the variable }}x \cr
& = \frac{3}{2}\left( {\sqrt {4{{\left( 5 \right)}^2} + 2{y^2}} } \right) - \frac{3}{2}\left( {\sqrt {4{{\left( 0 \right)}^2} + 2{y^2}} } \right) \cr
& {\text{simplifying}} \cr
& = \frac{3}{2}\left( {\sqrt {100 + 2{y^2}} } \right) - \frac{3}{2}\left( {\sqrt {2{y^2}} } \right) \cr
& = \frac{3}{2}\left( {\sqrt {100 + 2{y^2}} - \sqrt 2 y} \right) \cr} $$
