Answer
$(\ln 2)^2$
Work Step by Step
Re-arrange the given integral as follows:
$\int_{1}^{2} \int_{1}^{2}\dfrac{1}{xy} dy dx=(\dfrac{1}{2}) \ln (x)|_{1}^{2} (\dfrac{1}{y}) dy$
This implies that
$(\dfrac{1}{2}) \ln (x)|_{1}^{2} (\dfrac{1}{y}) dy=\ln (2)\int_{1}^{2} (\dfrac{dy}{y})$
Hence, $(\ln 2) [(\ln y)]_1^2=(\ln 2)^2$
