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6.6 Improper Integrals

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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

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Examples All three integrals are convergent.

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

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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

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

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