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MTH 252 Integral Calculus Chapter 8 – Principles of Integral Evaluation Section 8.8 – Improper Integrals Copyright © 2006 by Ron Wallace, all rights reserved.

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Definite Integrals where Assumptions? f(x) is continuous over [ a,b ] But what if … f(x) is only continuous over ( a,b ] f(x) is only continuous over [ a,b ) f(x) is only continuous over ( a,b ) f(x) is not continuous at c ( a,b ) a = b = Improper Integrals

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Examples Not continuous at x=0. Not continuous at x=1. Upper limit is infinite.

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Improper Integral over an Open Interval ab f(x) is not continuous at b. If f(x) is not continuous at a then… If f(x) is not continuous at a & b and a < c < b then…

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Improper Integrals Example Not continuous at x=0. Integration by Parts! L’Hôpital’s Rule

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Improper Integral over an Infinite Interval k

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Improper Integrals Example Upper limit is infinite. Divergent!

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Improper Integrals with a Point of Discontinuity

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Improper Integrals Example Not continuous at x=1. Divergent!

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Improper Integrals Example Not continuous at x=1. Warning! This is not correct … why? f(x) is NOT continuous over [ 0,3 ].

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