Answer
$$21$$
Work Step by Step
$$\eqalign{
& \int_1^2 {\int_4^9 {\frac{{3 + 5y}}{{\sqrt x }}} } dxdy \cr
& = \int_1^2 {\left[ {\int_4^9 {\frac{{3 + 5y}}{{\sqrt x }}} dx} \right]} dy \cr
& {\text{solve the inner integral}} \cr
& \int_4^9 {\frac{{3 + 5y}}{{\sqrt x }}} dx = \left( {3 + 5y} \right)\int_4^9 {\frac{1}{{\sqrt x }}} dx \cr
& = \left( {3 + 5y} \right)\int_4^9 {{x^{ - 1/2}}} dx \cr
& {\text{using the power rule }} \cr
& = \left( {3 + 5y} \right)\left( {\frac{{{x^{1/2}}}}{{1/2}}} \right)_4^9 \cr
& = 2\left( {3 + 5y} \right)\left( {\sqrt x } \right)_4^9 \cr
& {\text{evaluating the limits in the variable }}x \cr
& = 2\left( {3 + 5y} \right)\left( {\sqrt 9 - \sqrt 4 } \right) \cr
& {\text{simplifying}} \cr
& = 2\left( {3 + 5y} \right) \cr
& {\text{then}} \cr
& \int_1^2 {\int_4^9 {\frac{{3 + 5y}}{{\sqrt x }}} } dxdy = \int_1^2 {2\left( {3 + 5y} \right)} dy \cr
& {\text{integrating}} \cr
& = 2\left[ {3y + \frac{{5{y^2}}}{2}} \right]_1^2 \cr
& = 2\left[ {3\left( 2 \right) + \frac{{5{{\left( 2 \right)}^2}}}{2}} \right] - 2\left[ {3\left( 1 \right) + \frac{{5{{\left( 1 \right)}^2}}}{2}} \right] \cr
& = 32 - 11 \cr
& = 21 \cr} $$
