\displaystyle \frac{1}{2} + \int_{0.5}^1 \left( \int_0^{\frac{1}{2x}}1dy \right) dx = \frac{1}{2} + \int_{0.5}^1\frac{1}{2x} 
= \frac{1}{2} + \frac{1}{2} \left( \ln1 - \ln\left( \frac{1}{2} \right) \right) = \frac{1}{2} + \frac{1}{2}\ln2