अनन्त समाकल

अनंत समाकल (improper integral या infinite integral) निम्नलिखित प्रकार के होते हैं-

An improper integral of the first kind. The integral may need to be defined on an unbounded domain.

An improper Riemann integral of the second kind. The integral may fail to exist because of a vertical asymptote in the function.

$\lim _{b\to \infty }\int _{a}^{b}f(x)\,\mathrm {d} x,\qquad \lim _{a\to -\infty }\int _{a}^{b}f(x)\,\mathrm {d} x,$

or of the form

$\lim _{c\to b^{-}}\int _{a}^{c}f(x)\,\mathrm {d} x,\quad \lim _{c\to a^{+}}\int _{c}^{b}f(x)\,\mathrm {d} x,$

उदाहरण

$\int _{1}^{\infty }{\frac {1}{x^{2}}}\,\mathrm {d} x=\lim _{b\to \infty }\int _{1}^{b}{\frac {1}{x^{2}}}\,\mathrm {d} x=\lim _{b\to \infty }\left(-{\frac {1}{b}}+{\frac {1}{1}}\right)=1.$