6.6 Improper Integrals

Definition of an Improper Integral of Type 1 a)If exists for every number t ≥ a, then provided this limit exists (as a finite number). b)If exists for every number t ≤ b, then provided this limit exists (as a finite number). The improper integrals and are called convergent if the corresponding limit exists and divergent if the limit does not exist. c) If bothandare convergent, then we define

Examples All three integrals are convergent.

An example of a divergent integral: The general rule is the following:

Definition of an Improper Integral of Type 2 a)If f is continuous on [a, b) and is discontinuous at b, then if this limit exists (as a finite number). a)If f is continuous on (a, b] and is discontinuous at a, then if this limit exists (as a finite number). The improper integral is called convergent if the corresponding limit exists and divergent if the limit does not exist. c) If f has a discontinuity at c, where a < c < b, and both and are convergent, then we define

Example 1: This integral is divergent. Example 2: This integral is convergent.