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?

$$
\int_{\alpha\varepsilon}^{\alpha N} + \int_{\alpha N}^{\beta N} + \int_{\beta N}^{\beta\varepsilon} + \int_{\beta\varepsilon}^{\alpha\varepsilon} \quad = \quad \int_{\alpha\varepsilon}^{\alpha\varepsilon} = 0. 
$$
.
$$
\int_{\alpha\varepsilon}^{\alpha N} + \int_{\alpha N}^{\beta N} + \int_{\beta N}^{\beta\varepsilon} + \int_{\beta\varepsilon}^{\alpha\varepsilon}  = 0. 
$$
You can subtract an integral from both sides, and you can reverse $\text{“}{\pm}\text{''}$ by reversing the upper and lower bounds of integration.
