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Improper Integrals

Improper Integrals. Infinite Integrand Integrand is unbounded at some point on the interval of integration. Infinite Interval One or both of the limits of integration are infinite Lower limit of . Upper limit of . The Big Question. Is an improper integral convergent or divergent?

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Improper Integrals

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  1. Improper Integrals • Infinite Integrand • Integrand is unbounded at some point on the interval of integration. • Infinite Interval • One or both of the limits of integration are infinite • Lower limit of . • Upper limit of .

  2. The Big Question • Is an improper integral convergent or divergent? • Convergent • Has a sensible finite value. • Divergent • Doesn’t have a sensible finite value.

  3. Infinite Interval Consider the integral , where f is continuous for xa. If the limit exists and is finite, then Iconverges to L. Otherwise, Idiverges.

  4. Infinite Integrand Let the integral be improper either at a or at b. (The cases a =  and b =  are allowed). If either or exists and has finite value L, then Iconverges to L. Otherwise, Idiverges.

  5. Comparison Theorem • Let f and g be continuous functions and suppose that for all x a, 0  f (x)  g(x). • If converges, then so does , and • If diverges, then so does .

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