Improper Integrals

1
Improper Integrals

• Infinite Integrand
• 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?
• Convergent
• Has a sensible finite value.
• Divergent
• Doesnt 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
  .