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 interval(s).

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

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

or

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

where in each case one takes a limit in one of integration endpoints (Apostol 1967, §10.23). Of course, limits in both endpoints are also possible and this case is also considered as an improper integral.

By abuse of notation, improper integrals are often written symbolically just like standard definite integrals, perhaps with infinity among the limits of integration interval(s). 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.

The purpose of using improper integrals is that one is often 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 as an integrand or because one of the bounds of integration is infinite.