Intro to Analysis in-class_Part_43

Intro to Analysis in-class_Part_43 - 6.3 Improper Integrals...

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Unformatted text preview: 6.3 Improper Integrals 91 6.3 Improper Integrals Definition 6.3.1. Roughly, an improper integral is an integral which can only be defined on a domain by taking a limit of its values on subdomains. Example 6.3.1. f ( x ) = x α has an integrable singularity at x = 0 iff α >- 1 and an integrable singularity at ∞ iff α <- 1. Solution. Z t s x α dx = h x α +1 α +1 i t s , α 6 =- 1 , [log x ] t s , α =- 1 , = t α +1- s α +1 α +1 , α 6 =- 1 , log t log s , α =- 1 . Let α >- 1, so α + 1 > 0. Then we have the improper integrals Z t x α dx = lim s → Z t s x α dx = t α +1 α + 1 , Z ∞ s x α dx = lim t →∞ Z t s x α dx = ∞ . Let α <- 1, so α + 1 < 0. Then we have the improper integrals Z t x α dx = lim s → Z t s x α dx = ∞ , Z ∞ s x α dx = lim t →∞ Z t s x α dx =- s α +1 α + 1 . If α =- 1, then log t log s diverges as s → 0 or as t → ∞ , so neither improper integral exists....
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This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

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Intro to Analysis in-class_Part_43 - 6.3 Improper Integrals...

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