Improper integral

Description

The improper integral (uneigentliches Integral) is used if the the upper or lower bound are singularities or lie at infinity.

Definition

One-sided: Let $a,b\in ]-\infty ,\infty ]$, $a<b$ (with $b$ possibly infinity). Then the improper integral is defined as follows: ${\int }_a^{b}f(x)dx:={\lim }_{\beta \nearrow b}{\int }_a^{\beta }f(x)dx$ if defined. Two sided: Let $a,b\in [-\infty ,\infty ]$, $a<b$ (with $a,b$ possibly infinity). Let $m\in ]a,b[$. Then the improper integral is defined as follows: ${\int }_a^{b}f(x)dx:={\int }_a^{m}f(x)dx+{\int }_m^{b}f(x)dx={\lim }_{\alpha \searrow a}{\int }_{\alpha }^{m}f(x)dx+{\lim }_{\beta \nearrow b}{\int }_m^{\beta }f(x)dx$ if defined.

Properties

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Examples

Example