I am working my way through Mark Homes Introduction to Perturbation Methods by myself. I am having trouble solving problem 1.8(c).
I believe that I'm supposed to split the integral by introducing a small interval $\delta$, $0 \approx \delta \ll 0.5$ since the term $x(x-1)$ is symmetric about the y-axis at $x=0.5$, reducing the term in the integral to $\frac{1}{\varepsilon - x}$. However, I have the solution from Holmes' public answer sheet and that approach doesn't seem to be part of the solution.
I believe that the second $dx$ in the equation is a typo, though I have not found the errata for this textbook.