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Concept
Improper integral
Summary
In mathematical analysis, an improper integral is an extension of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context of Riemann integrals (or, equivalently, Darboux integrals), this typically involves unboundedness, either of the set over which the integral is taken or of the integrand (the function being integrated), or both. It may also involve bounded but not closed sets or bounded but not continuous functions. While an improper integral is typically written symbolically just like a standard definite integral, it actually represents a limit of a definite integral or a sum of such limits; thus improper integrals are said to converge or diverge. If a regular definite integral (which may retronymically be called a proper integral) is worked out as if it is improper, the same answer will result.
In the simplest case of a real-valued function of a single variable integrated in the sense of Riemann (or Darboux) over a single interval, improper integrals may be in any of the following forms:
where is undefined or discontinuous somewhere on
The first three forms are improper because the integrals are taken over an unbounded interval. (They may be improper for other reasons, as well, as explained below.) Such an integral is sometimes described as being of the "first" type or kind if the integrand otherwise satisfies the assumptions of integration. Integrals in the fourth form that are improper because has a vertical asymptote somewhere on the interval may be described as being of the "second" type or kind. Integrals that combine aspects of both types are sometimes described as being of the "third" type or kind.
In each case above, the improper integral must be rewritten using one or more limits, depending on what is causing the integral to be improper. For example, in case 1, if is continuous on the entire interval , then
The limit on the right is taken to be the definition of the integral notation on the left.
Official source
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