Intro to Analysis in-class_Part_43

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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online