What is an improper integral?

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What is an improper integral?

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Answer 1 · 2015-04-14T17:31:09

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

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

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

Examples:

$\int_{0}^{1} ln x d x$ $int_0^1 lnx dx$ $sssssssssss$ $color(white)"sssssssssss"$ integrand not defined at $0$ $0$

$\int_{5}^{7} \frac{1}{x^{2} - 25} d x$ $int_5^7 1/(x^2-25) dx$ $ssssss$ $color(white)"ssssss"$ integrand not defined at $5$ $5$

$\int_{1}^{\infty} \frac{1}{x^{2}} d x$ $int_1^oo 1/x^2 dx$ $sssssssssss$ $color(white)"sssssssssss"$ interval has no right endpoint

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What is an improper integral?

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