Improper Integrals

GRE Math Subject · Axiom Academy

Integrals with infinite limits or unbounded integrands -- convergence, comparison, and the Gamma function What Makes an Integral Improper? An integral is improper if at least one of these holds: Type I: One or both limits of integration are infinite. Type II: The integrand has a discontinuity (vertical asymptote) within the interval of integration. In either case, we define the integral as a limit . If the limit exists and is finite, the integral converges ; otherwise it diverges . Type I: Infinite Limits of Integration For both limits infinite, split at any finite point c: Both integrals on the right must converge independently. Type II: Discontinuous Integrand Definition: If f has a vertical asymptote at x = c where a c b: If the discontinuity is at x = a: If the discontinuity is at x = b: Common GRE trap: Blindly applying FTC to gives 0, but the integrand has a discontinuity at x = 0. You must split at 0 and check each piece. Both diverge, so the integral diverges. The p-Integral (Critical Reference) Converges if and only if p > 1. When it converges, the value is . Converges if and only if p < 1. These two results are mirror images -- memorize both. The GRE frequently tests the boundary case p = 1 (harmonic-style integrals). Comparison Test for Improper Integrals Direct Comparison: Suppose 0 f(x) g(x) for all x a. If converges, then also converges. If diverges, then also diverges. Limit Comparison: If where 0 < L < , then and either both converge or both diverge.

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