Improper Integrals

Objective

• Evaluate an integral over an infinite interval.
• Evaluate an integral over a closed interval with an infinite discontinuity within the interval
• Use the comparison theorem to determine whether a definite integral is convergent

Integrating over an Infinite Interval

• When integrating over an infinite interval, it is reasonable that we look at the behavior of the integral as the limit of integration approaches or

Definition
Integrating over an Infinite Interval

• Let be continuous over the interval .

• Let be continuous over the interval .

For both cases above, if the limit exists, then the improper integral converges.
If the limit does not exist, the the improper integral diverges.

• Let be continuous over the interval and be any real number over the domain of .
  

If both and converges, then converges.
If either one or both of these two diverges, then diverges

Integrating over a Discontinuous Integrand

• Consider the integral , where is continuous over but discontinuous at .
  • To deal with this, we consider that is continuous over .
    • However, we should also know that must satisfy .
  • Therefore, for us to integrate this, we consider the one-sided limit as approaches from the left.
• We can do similar approaches for any integral with a discontinuous integrand.

Definition
Integrating a Discontinuous Integrand

• Let be continuous over .
• Let be continuous over .

In each case, if the limit exists, then the improper integral converges.
If the limit does not exist, then the improper integral diverges.

• Let be a point in . If is continuous over [a, b] except at point , then

If both integrals and converge, then converges.
If either one or both of them diverges, then diverges.

Comparison Theorem

• By comparing two similar functions and , we can simply compare them without evaluating if an improper integral diverges or converges.

Definition
Comparison Test for Improper Integrals

Suppose that for all , .

• Case 1: If :
  • Then, this indicates that must diverge as well.
• Case 2: converges to some value :
  • Then, this means that must converge to some other value that is less than or equal to .