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.

Famous quotes containing the words improper and/or integral:

“He never does a proper thing without giving an improper reason for it.”
—George Bernard Shaw (1856–1950)

“... no one who has not been an integral part of a slaveholding community, can have any idea of its abominations.... even were slavery no curse to its victims, the exercise of arbitrary power works such fearful ruin upon the hearts of slaveholders, that I should feel impelled to labor and pray for its overthrow with my last energies and latest breath.”
—Angelina Grimké (1805–1879)