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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 has an integrable singularity at x = 0 iff >- 1 and an integrable singularity at iff <- 1. Solution. Z t s x dx = h x +1 +1 i t s , 6 =- 1 , [log x ] t s , =- 1 , = t +1- s +1 +1 , 6 =- 1 , log t log s , =- 1 . Let >- 1, so + 1 > 0. Then we have the improper integrals Z t x dx = lim s Z t s x dx = t +1 + 1 , Z s x dx = lim t Z t s x dx = . Let <- 1, so + 1 < 0. Then we have the improper integrals Z t x dx = lim s Z t s x dx = , Z s x dx = lim t Z t s x dx =- s +1 + 1 . If =- 1, then log t log s diverges as s 0 or as t , so neither improper integral exists....
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