$\Large \displaystyle \int_0^1 \int_{\frac{e^x}{12}}^\frac{e^{\sqrt{x}}}{12} ~\dfrac{1}{\ln(12y)}~dy~dx$
Trying to integrate it in this order is a mess. a chocolate mess.
So let's flip the order of integration, especially as $f(x,y)$ doesn't depend on $x$ at all
$\Large \displaystyle \int_{\frac{1}{12}}^{\frac{e^1}{12}} \int_{\ln(12y)^2}^{\ln(12y)}~\dfrac{1}{\ln(12y)}~d x~dy = $
$\Large \displaystyle \int_{\frac{1}{12}}^{\frac{e^1}{12}}~\dfrac{\ln (12 y)-\ln ^2(12 y)}{\ln (12 y)}~dy =$
$\Large \displaystyle \int_{\frac{1}{12}}^{\frac{e^1}{12}}~1 - \ln(12y)~dy =$
$\Large \left . 2 y-y \log (12 y) \right|_\frac{1}{12}^{\frac{e^1}{12}} = \dfrac{1}{12} (e-2) \approx 0.0598568$