improper-integrals-type-i-part-ii-infinite-limits-example-xex-ap-calculus-ab-bc-limits

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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. But that conceals the limiting process. By using the more advanced Lebesgue integral, rather than the Riemann integral, one can in some cases get an answer without taking a limit of standard definite integrals.

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 (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.- source wikipedia

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