Answer
$$\frac{9}{5} + \frac{7}{2}\ln 2$$
Work Step by Step
$$\eqalign{
& \int_3^4 {\int_1^2 {\left( {\frac{{6x}}{5} + \frac{y}{x}} \right)} } dxdy \cr
& = \int_3^4 {\left[ {\int_1^2 {\left( {\frac{{6x}}{5} + \frac{y}{x}} \right)} dx} \right]} dy \cr
& {\text{solve the inner integral}} \cr
& \int_1^2 {\left( {\frac{{6x}}{5} + \frac{y}{x}} \right)} dx = \left( {\frac{{3{x^2}}}{5} + y\ln \left| x \right|} \right)_1^2 \cr
& {\text{evaluate limits for }}x \cr
& = \left( {\frac{{3{{\left( 2 \right)}^2}}}{5} + y\ln \left| 2 \right|} \right) - \left( {\frac{{3{{\left( 1 \right)}^2}}}{5} + y\ln \left| 1 \right|} \right) \cr
& = \left( {\frac{{12}}{5} + y\ln 2} \right) - \left( {\frac{3}{5} + y\left( 0 \right)} \right) \cr
& = \frac{9}{5} + y\ln 2 \cr
& {\text{then}} \cr
& \int_3^4 {\int_1^2 {\left( {\frac{{6x}}{5} + \frac{y}{x}} \right)} } dxdy = \int_3^4 {\left( {\frac{9}{5} + y\ln 2} \right)} dy \cr
& {\text{integrating}} \cr
& \left[ {\frac{9}{5}y + \frac{{{y^2}}}{2}\left( {\ln 2} \right)} \right]_3^4 \cr
& = \left( {\frac{9}{5}\left( 4 \right) + \frac{{{{\left( 4 \right)}^2}}}{2}\left( {\ln 2} \right)} \right) - \left( {\frac{9}{5}\left( 3 \right) + \frac{{{{\left( 3 \right)}^2}}}{2}\left( {\ln 2} \right)} \right) \cr
& = \frac{{36}}{5} + 8\ln 2 - \frac{{27}}{5} - \frac{9}{2}\ln 2 \cr
& = \frac{9}{5} + \frac{7}{2}\ln 2 \cr} $$
