1. 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 .

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

3. 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.

4. 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.

5. 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