Improper integral

In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number or positive or negative infinity; or in some instances as both endpoints approach limits. Such an integral is often written symbolically just like a standard definite integral, in some cases with infinity as a limit of integration. An improper integral of the first kind. The integral may need to be defined on an unbounded domain. An improper Riemann integral of the second kind. The integral may fail to exist because of a vertical asymptote in the function.

Specifically, an improper integral is a limit of the form:

$\lim _{b\to \infty }\int _{a}^{b}f(x)\,dx,\quad \lim _{a\to -\infty }\int _{a}^{b}f(x)\,dx$ or

$\lim _{c\to b^{-}}\int _{a}^{c}f(x)\ dx,\quad \lim _{c\to a^{+}}\int _{c}^{b}f(x)\ dx$ in which one takes a limit in one or the other (or sometimes both) endpoints (Apostol 1967, §10.23).

By abuse of notation, improper integrals are often written symbolically just like standard definite integrals, perhaps with infinity among the limits of integration. When the definite integral exists (in the sense of either the Riemann integral or the more powerful Lebesgue integral), this ambiguity is resolved as both the proper and improper integral will coincide in value.

Often one is able to compute values for improper integrals, even when the function is not integrable in the conventional sense (as a Riemann integral, for instance) because of a singularity in the function or because one of the bounds of integration is infinite.