Answer
$\ln 2$
Work Step by Step
Re-arrange the given integral as follows:
$\int_{1}^{4} [\dfrac{\ln x}{xy} dx]_{1}^{e}=\dfrac{1}{4} \int_{1}^{4}\dfrac{\ln x}{x} dx$
Plug $ \ln x=p \implies dp=\dfrac{dx}{x}$
This implies that
$(\dfrac{1}{4}) \int_{1}^{4} \dfrac{p^2}{2}|_1^e =(\dfrac{1}{4}) \dfrac{\ln ^2 x}{2}]_1^e$
Hence, $\int_1^{4} \dfrac{dy}{2y} =\dfrac{1}{2} \ln (y)|_1^4$
or, $\dfrac{\ln 4}{2}=\ln 2$
