Improper Integral

Definition

Open Interval

When the interval, subject to integral, is an open. $(a,b]$ or $[a,b)$

Suppose $f:(a,b]\to \mathbb{R}$, and $\forall c\in (a,b),f\in \mathcal{R}[c,b]$, then the improper integral of $f$ on a $(a,b]$ is defined as ${\int }_a^{b}f(x)dx={\lim }_{c\to a^{+}}{\int }_c^{b}f(x)dx$

Suppose $f:[a,b)\to \mathbb{R}$, and $\forall c\in (a,b),f\in \mathcal{R}[c,b]$, then the improper integral of $f$ on a $[a,b)$ is defined as ${\int }_a^{b}f(x)dx={\lim }_{c\to b^{-}}{\int }_a^{c}f(x)dx$

If there exists a limit of the expression, we call $f$ is improper integrable.

If $f:[a,b]-\{c\}\to \mathbb{R}$ is improper integrable on both $[a,c)$ and $(c,b]$, then $f$ is improper integrable on a $[a,b]$ and define ${\int }_a^{b}f(x)dx={\lim }_{p\to c^{-}}{\int }_a^{p}f(x)dx+{\lim }_{q\to c^{+}}{\int }_q^{b}f(x)dx$

Unbounded Interval

When the interval, subject to integral, is an unbounded. $(-\infty ,b]$ or $[a,\infty )$

Suppose $f:[a,\infty )\to \mathbb{R}$, and $\forall c\in \mathbb{R},a<c\Rightarrow f\in \mathcal{R}[a,c]]$, then the improper integral of $f$ on a $[a,\infty )$ is defined as ${\int }_a^{\infty }f(x)dx={\lim }_{c\to \infty }{\int }_a^{c}f(x)dx$

Suppose $f:(-\infty ,b]\to \mathbb{R}$, and $\forall c\in \mathbb{R},c<b\Rightarrow f\in \mathcal{R}[c,b]]$, then the improper integral of $f$ on a $(-\infty ,b]$ is ${\int }_{-\infty }^{b}f(x)dx={\lim }_{c\to -\infty }{\int }_c^{b}f(x)dx$

If there exists a limit of the expression, we call $f$ is improper integrable.

If $f$ is improper integrable on both $(-\infty ,p]$ and $[p,\infty )$, then $f$ is improper integrable on a $\mathbb{R}$ and define ${\int }_{-\infty }^{\infty }f(x)dx={\int }_{-\infty }^{p}f(x)dx+{\int }_p^{\infty }f(x)dx$