Overview of Improper IntegralsMAT 104 – Frank Swenton, Summer 2000Definitions•Aproper integralis a definite integral where the interval isfiniteand the integrand isdefinedandcontinuousat all points in the interval.– Proper integrals always converge, that is, always give a finite area•Thetrouble spotsof a definite integral are the points in the interval of integration that make itan improper integral, i.e., keep it from being proper. They are of two types:a. Points where the integrand is undefined or discontinuousb.∞and-∞are always trouble spots when they appear as limits of integration•Asimple improper integralis an improper integral with only one trouble spot, that trouble spotbeing at an endpoint of the interval. Simple improper integrals are defined to be the appropriatelimits of proper integrals, e.g.:Z101xdx=limε→0+Z1ε1xdx– If the limit exists as a real number, then the simple improper integral is calledconvergent.– If the limit doesn’t exist as a real number, the simple improper integral is calleddivergent.Dealing with improper integrals•First step: always locatealltrouble spots and split the integral intosimpleimproper integrals, thendeal with the pieces individually.
