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 โˆž or โˆ’โˆž 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, ยง10.23). 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.

Read more about Improper Integral:  Examples, Convergence of The Integral, Types of Integrals, Improper Riemann Integrals and Lebesgue Integrals, Singularities, Cauchy Principal Value, Summability, Bibliography

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)

    “Painting myself for others, I have painted my inward self with colors clearer than my original ones. I have no more made my book than my book has made me—a book consubstantial with its author, concerned with my own self, an integral part of my life; not concerned with some third-hand, extraneous purpose, like all other books.”
    —Michel de Montaigne (1533–1592)