Improper Integral

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.

Specifically, an improper integral is a limit of the form

or of the form

$\lim_{c\to b^-} \int_a^cf(x)\, \mathrm{d}x,\quad \lim_{c\to a^+} \int_c^bf(x)\, \mathrm{d}x,$

in which one takes a limit in one or the other (or sometimes both) endpoints (Apostol 1967, §10.23). Integrals are also improper if the integrand is undefined at an interior point of the domain of integration, or at multiple such points.

It is often necessary to use improper integrals in order to compute a value for integrals which may not exist in the conventional sense (as a Riemann integral, for instance) because of a singularity in the function, or an infinite endpoint of the domain of integration.

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