Answer
$$\frac{{26}}{3}$$
Work Step by Step
$$\eqalign{
& \int_1^3 {\int_1^{{e^x}} {\frac{x}{y}} } dydx \cr
& = \int_1^3 {\left[ {\int_1^{{e^x}} {\frac{x}{y}} dy} \right]} dx \cr
& {\text{solve the inner integral}}{\text{, treat }}x{\text{ as a constant}} \cr
& \int_1^{{e^x}} {\frac{x}{y}} dy \cr
& = \left[ {x\ln \left| y \right|} \right]_1^{{e^x}} \cr
& {\text{evaluating the limits in the variable }}y \cr
& = \left[ {x\ln \left| {{e^x}} \right| - x\ln \left| 1 \right|} \right] \cr
& {\text{simplifying}} \cr
& = \left[ {x\left( x \right) - x\left( 0 \right)} \right] \cr
& = {x^2} \cr
& \cr
& \int_1^3 {\left[ {\int_1^{{e^x}} {\frac{x}{y}} dy} \right]} dx = \int_1^3 {{x^2}} dx \cr
& {\text{integrating by using the power rule}} \cr
& = \left( {\frac{{{x^3}}}{3}} \right)_1^3 \cr
& {\text{evaluate}} \cr
& = \frac{{{{\left( 3 \right)}^3}}}{3} - \frac{{{{\left( 1 \right)}^3}}}{3} \cr
& = 9 - \frac{1}{3} \cr
& = \frac{{26}}{3} \cr} $$
