Improper Integrals

A Problem With Riemann Integrals

Explanation that Riemann integrals are not defined when the interval is unbounded but can often be found using limits. Mention of the alternative Lebesgue integral.

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Improper Integrals of the First Kind

Using limits to evaluate improper integrals with unbounded limits of integration. Includes examples of integrating 1/(xp) from 1 to infinity and integrating cos(x) from 0 to infinity.

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Improper Integrals of the Second Kind

Using limits to evaluate improper integrals involving functions that are unbounded over the specified limits of integration. Includes example of integrating 1/(xp) between 0 and 1.

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The Comparison Test

Definition of monotone bounded limits, the squeezing lemma for limits and improper integrals, and the comparison test for convergence of improper integrals.

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Improper Integrals

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The Comparison Test for determining convergence or divergence of improper integrals, with discussion and examples.

An integral with an infinite upper limit of integration to be evaluated.

Determining whether twenty-two different improper integrals are convergent or divergent, and evaluating the limits of six integrals using the Fundamental Theorem.

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Determining whether an improper integral converges or diverges.

Two questions which involve determining whether an improper integral will converge or diverge.