# Showing that a finction is not integrable

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• Aug 16th 2011, 11:12 AM
mgarson
Showing that a finction is not integrable
Hello!

I've got what looks like a simple problem but I just can't get it right.

Here it goes: Show that $\displaystyle f(x)=d/dx (x^2\sin x^{-3}), x>0$, is not in $\displaystyle L^1(0,1)$.

My first thought was to simply compute the integral
$\displaystyle \int_{0}^{1}|f(x)|dx$ and show that it diverges - however I had no luck doing that.

Then I thought that I might simply estimate it from below with something that I know is infinite but no luck there either, all I got was:
$\displaystyle \int_{0}^{1}|2x\sin x^{-3}-3x^{-2}\cos x^{-3}|dx \geq \int_{0}^{1}(|2x\sin x^{-3}|-|3x^{-2}\cos x^{-3}|)dx$

Thanks!
• Aug 16th 2011, 12:49 PM
girdav
Re: Showing that a finction is not integrable
You don't need to compute the integral. Just notice that $\displaystyle \sin (x^{-3})\overset{0}{\sim} x^{-3}$ and the integral $\displaystyle \int_0^1\frac 1xdx$ is divergent.