Showing that a finction is not integrable

**mgarson** · August 16th 2011, 11:12 AM

Hello!

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

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

My first thought was to simply compute the integral
$\int_{0}^{1}|f(x)|dx$ 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:
$\int_{0}^{1}|2x\sin x^{-3}-3x^{-2}\cos x^{-3}|dx \geq \int_{0}^{1}(|2x\sin x^{-3}|-|3x^{-2}\cos x^{-3}|)dx$

Thanks!

**girdav** · August 16th 2011, 12:49 PM

You don't need to compute the integral. Just notice that $\sin (x^{-3})\overset{0}{\sim} x^{-3}$ and the integral $\int_0^1\frac 1xdx$ is divergent.

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