Question

What is an improper integral?

Answer 1

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

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

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 lnx dx` `color(white)"sssssssssss"` integrand not defined at `0`

`int_5^7 1/(x^2-25) dx` `color(white)"ssssss"` integrand not defined at `5`

`int_1^oo 1/x^2 dx` `color(white)"sssssssssss"` interval has no right endpoint