(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find a real function f on (0,1] and integrable on [c,1] for every c>0, such that [tex]\int^{1}_{0} f(x)dx[/tex] exists but [tex]\int^{1}_{0}|f(x)|dx [/tex] fails to exist

2. Relevant equations

3. The attempt at a solution

I think such function should behavior like the sequence 1-1/2+1/3-1/4+....+(-1)^n-1 1/n+.....so I came up to something like [tex]\frac{1}{x^2}[/tex] sin([tex]\frac{1}{x}[/tex])

But this function is not integrable on [0,1].

Moreover, I find that I don't know how to deal with the integration with |f(x)| instead. for example, I cannot find a way to integrate [tex]\int{\frac{1}{x^2} |sin(\frac{1}{x})|} dx[/tex] (can it be solved?)

Perhaps the function I need may have a totally different form....

Any hint?

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# Homework Help: Improper integration

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