Sunday, 6 October 2013

Improper Integrals

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
\lim_{b\to\infty} \int_a^bf(x)\, \mathrm{d}x, \qquad \lim_{a\to -\infty} \int_a^bf(x)\, \mathrm{d}x,
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 endpoints.
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