# What is an improper integral?

Apr 14, 2015

The definite integral over interval $\left[a , b\right]$ of $f$ is initially defined For a function $f$ that includes $\left[a , b\right]$ in its domain.

That is: we start with a function $f$ that is defined for all $x \in \left[a , b\right]$

Improper integrals extend the initial definition by allowing
$a$, or $b$, or both to be outside the domain of $f$ (but on the 'edge' so we can look for limits)
or for the interval to lack left and/or right endpoints (infinite intervals).

Examples:

${\int}_{0}^{1} \ln x \mathrm{dx}$ $\textcolor{w h i t e}{\text{sssssssssss}}$ integrand not defined at $0$

${\int}_{5}^{7} \frac{1}{{x}^{2} - 25} \mathrm{dx}$ $\textcolor{w h i t e}{\text{ssssss}}$ integrand not defined at $5$

${\int}_{1}^{\infty} \frac{1}{x} ^ 2 \mathrm{dx}$ $\textcolor{w h i t e}{\text{sssssssssss}}$ interval has no right endpoint