Switching limits of integration The solution in my textbook wrote $$\int_{\alpha \epsilon}^{\alpha N} \frac{f(u)}{u} \, du-\int_{\beta \epsilon}^{\beta N} \frac{f(u)}{u} \, du = \int_{\alpha \epsilon}^{\beta \epsilon} \frac{f(u)}{u} \, du-\int_{\alpha N}^{\beta N} \frac{f(u)}{u} \, du.$$ How can the limits of integration be switched like that?

If you write the equation with sums instead of differences, it reads:
$$
    \int_{\alpha \epsilon}^{\alpha N} \dfrac{f(u)}{u}du
+\int_{\alpha N}^{\beta N} \dfrac{f(u)}{u}du
= \int_{\alpha \epsilon}^{\beta \epsilon} \dfrac{f(u)}{u}du
   +\int_{\beta \epsilon}^{\beta N} \dfrac{f(u)}{u}du 
$$
Now if you use the additive rule for integrals over adjacent intervals, you see that each of these is equal to 
$$
    \int_{\alpha \epsilon}^{\beta N} \dfrac{f(u)}{u}du
$$
