F is integrable but f^2 is not integrable

[pulin816]

Homework Statement

Give an example of a sequence of integrable functions f:R -> R such that f is integrable but f^2 is not integrable. Prove your results.

The Attempt at a Solution

would 1/sqrt(x) for x ∈ [0,1] work? any other suggestions?
how would you prove that its does work? woudl you just plug it in and show it doesn't work or is there something more to it?

Thanks! I really appreciate your help!

 

[Dick]

Sure f(x)=1/sqrt(x) on [0,1] works. Just integrate it. It's improper, but that shouldn't be a problem.

 

[pulin816]

super. I thought proofs had to be more complicated, somehow.
Is there any theorem that explains integrability?

 

[Dick]

super. I thought proofs had to be more complicated, somehow.
Is there any theorem that explains integrability?

There's a lot of theorems about integrability. There's no one that explains everything. If you can do it as an elementary integral, like this one, you don't need anything complicated. Just a little imagination.