Answer
$\displaystyle \frac{\ln 3(\ln 5)^{2}}{2}$
Work Step by Step
Let $g(x)=\displaystyle \frac{1}{x}, \quad h(y)=\frac{\ln y}{y}.$
Apply formula (5), $\displaystyle \iint_{R}g(x)\cdot h(y)dA=\int_{a}^{b}g(x)dx\cdot\int_{c}^{d}h(y)dy$
$\displaystyle \int_{1}^{3}\frac{1}{x}dx=[\ln|x|]_{1}^{3}=\ln 3-\ln 1=\ln 3$
$\displaystyle \int_{1}^{5}\frac{\ln y}{y}dy=\left[\begin{array}{l}
u=\ln y\\
du=\frac{dy}{y}
\end{array}\right]=\int_{\ln 1}^{\ln 5}udu=[\frac{u^{2}}{2}]_{0}^{\ln 5}=\frac{(\ln 5)^{2}}{2}$
So,
$\displaystyle \int_{1}^{3}\int_{1}^{5}\frac{\ln y}{xy}dydx=\int_{1}^{3}\frac{1}{x}dx\int_{1}^{5}\frac{\ln y}{y}dy$
$=\displaystyle \frac{\ln 3(\ln 5)^{2}}{2}$
