Improper Integrals

[ecoli]

I have a question about improper integrals. Say you have a function with an asymptote at, say x=5. But you only have to integrate until x=4. Is is still considered an improper integral?

[Dave]

I always thought that an improper integral is one defined with one or both limits set to [imath]\infty[/imath]? (i.e. [imath]\int_a^{\infty} f \, dx[/imath] or whatever).

[ecoli]

not necessarily. An integral can also be improper, such as the graph ln(x), there is an asympote at x=0 so it's improper. But is you integrate from, say 1 to 2, is it still considerexd improper?

[Dave]

Well, no. That's certainly a definite integral.

 

I've never been taught improper integrals explicitly so I'm taking my definition from MathWorld.

[TD]

@ Dave, Mathworld is correct but it included these kind of improper integrals.

An improper integral is a definite integral that has either or both limits infinite or an integrand that approaches infinity at one or more points in the range of integration.

The bold part includes the integrands with, for example, zeroes of the denominator so vertical asymptotes.

 

@ ecoli: no, the integral isn't considered as improper then

[Dave]

Fairy fluff. I need to learn to read

[ecoli]

What for? You're a mathematician.

[Dave]

Good point