What is "improper integral"

n. (context mathematics English) An integral where at least one of the endpoints is taken as a limit, either to a specific number or to infinity.

In calculus, 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 ∞ or  − ∞ or, in some cases, as both endpoints approach limits. Such an integral is often written symbolically just like a standard definite integral, perhaps with infinity as a limit of integration.

Specifically, an improper integral is a limit of the form

lim∫f(x) dx,   lim∫f(x) dx, 
or of the form

lim∫f(x) dx,  lim∫f(x) dx, 
in which one takes a limit in one or the other (or sometimes both) endpoints . When a function is undefined at finitely many interior points of an interval, the improper integral over the interval is defined as the sum of the improper integrals over the intervals between these points.

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 advanced 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 poor behavior at infinity. Such integrals are often termed "properly improper", as they cannot be computed as a proper integral.