Computing Two-Step Improper Integrals - Expii

We now understand "one-step" improper integrals: those with a single infinite endpoint (but no infinite singularity), or a single infinite singularity at a finite endpoint (but no infinite endpoint). We now discuss "two-step" improper integrals, which can be broken into two simpler, one-step improper integrals. For instance, if the integrand \(f(x)\) has a single infinite singularity at an interior point \(c\in (a,b)\) (where interval \([a,b]\) is assumed bounded), then the improper integral \(\int_a^b f(x)\,dx\) is defined as the sum of the one-step improper integrals \(\int_a^c f(x)\,dx\) and \(\int_{c}^b f(x)\,dx\). Similarly, if each endpoint \(a,b\) is "bad" (either infinite, or a finite point of infinite singularity) for \(f\), but \(f\) has no interior infinite singularities, then \(\int_a^b f(x)\,dx\) can be defined as the sum of the one-step improper integrals \(\int_a^c f(x)\,dx\) and \(\int_c^b f(x)\,dx\), for any choice of interior point \(c\in (a,b)\).