Improper integral

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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.^[1] 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.^[2][1] 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:

1. ${\displaystyle \int _{a}^{\infty }f(x)\,dx}$
2. ${\displaystyle \int _{-\infty }^{b}f(x)\,dx}$
3. ${\displaystyle \int _{-\infty }^{\infty }f(x)\,dx}$
4. ${\displaystyle \int _{a}^{b}f(x)\,dx}$, where ${\displaystyle f(x)}$ is undefined or discontinuous somewhere on ${\displaystyle [a,b]}$

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.^[2] Integrals in the fourth form that are improper because ${\displaystyle f(x)}$ has a vertical asymptote somewhere on the interval ${\displaystyle [a,b]}$ may be described as being of the "second" type or kind.^[2] Integrals that combine aspects of both types are sometimes described as being of the "third" type or kind.^[2]

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 ${\displaystyle f(x)}$ is continuous on the entire interval ${\displaystyle [a,\infty )}$, then

${\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{b\to \infty }\int _{a}^{b}f(x)\,dx.}$

The limit on the right is taken to be the definition of the integral notation on the left.

If ${\displaystyle f(x)}$ is only continuous on ${\displaystyle (a,\infty )}$ and not at ${\displaystyle a}$ itself, then typically this is rewritten as

${\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{t\to a^{+}}\int _{t}^{c}f(x)\,dx+\lim _{b\to \infty }\int _{c}^{b}f(x)\,dx,}$

for any choice of ${\displaystyle c>a}$. Here both limits must converge to a finite value for the improper integral to be said to converge. This requirement avoids the ambiguous case of adding positive and negative infinities (i.e., the "${\displaystyle \infty -\infty }$" indeterminate form). Alternatively, an iterated limit could be used or a single limit based on the Cauchy principal value.

If ${\displaystyle f(x)}$ is continuous on ${\displaystyle [a,d)}$ and ${\displaystyle (d,\infty )}$, with a discontinuity of any kind at ${\displaystyle d}$, then

${\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{t\to d^{-}}\int _{a}^{t}f(x)\,dx+\lim _{u\to d^{+}}\int _{u}^{c}f(x)\,dx+\lim _{b\to \infty }\int _{c}^{b}f(x)\,dx,}$

for any choice of ${\displaystyle c>d}$. The previous remarks about indeterminate forms, iterated limits, and the Cauchy principal value also apply here.

The function ${\displaystyle f(x)}$ can have more discontinuities, in which case even more limits would be required (or a more complicated principal value expression).

Cases 2–4 are handled similarly. See the examples below.

Improper integrals can also be evaluated in the context of complex numbers, in higher dimensions, and in other theoretical frameworks such as Lebesgue integration or Henstock–Kurzweil integration. Integrals that are considered improper in one framework may not be in others.