Improper Integrals Infinite Integrand Infinite Interval Integrand is unbounded at some point on the interval of integration. Infinite Interval One or both of the limits of integration are infinite Lower limit of . Upper limit of .

The Big Question Is an improper integral convergent or divergent? Has a sensible finite value. Divergent Doesn’t have a sensible finite value.

Infinite Interval Consider the integral , where f is continuous for x  a. If the limit exists and is finite, then I converges to L. Otherwise, I diverges.

Infinite Integrand Let the integral be improper either at a or at b. (The cases a =  and b =  are allowed). If either or exists and has finite value L, then I converges to L. Otherwise, I diverges.

Comparison Theorem Let f and g be continuous functions and suppose that for all x  a, 0  f (x)  g(x). If converges, then so does , and If diverges, then so does .