Expii

# 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 $$c\in (a,b)$$.