6.14 Evaluating Improper Integrals

This topic is BC only. An improper integral is a definite integral in which at least one limit of integration is infinite or the integrand becomes unbounded somewhere on the interval. Because a definite integral is defined over a finite interval with a bounded integrand, improper integrals are evaluated by rewriting them as limits. If the resulting limit exists and is finite, the improper integral converges. If the limit does not exist or is infinite, the improper integral diverges.

You should treat improper integrals as limit problems first and integral problems second. If the interval goes to infinity, replace the infinite bound with a variable such as b and take a limit as ${b \to \infty}$. If the function has a vertical asymptote inside or at an endpoint of the interval, split the integral if needed and use limits near the problematic point. This topic brings together limit reasoning and integration, and it reinforces that not every infinite-looking accumulation gives a finite result.