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Given the function

$$f(x)= \cases{ x\sin\Big(\frac{1}{x}\Big) & if $x\neq 0$ \\ 0 & if $x=0$} $$

Find $$\int\limits_{-\infty}^\infty \frac{1}{f(x)} dx $$

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    I partly fixed up the TeX, but the question does not make full sense, since $f(x)$ is not mentioned in the integral. Can you (by editing or leaving a comment) say what the question really is?2012-08-26
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    Is there any particular reason to believe that it converges? The integrand is $$1+O\left(\frac{1}{x^2}\right)$$ as $x\to\pm\infty$, which already shows that the integral does not converge, much less its highly singular behavior near the origin.2012-08-26
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    @sos440 it doesn't converge. You should write an answer.2012-08-26
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    @JenniferDylan, I suspect that the questioner might have made a mistake. So I'm going to wait for a while to see if it is true. It won't be too late then to post an answer.2012-08-26
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    Someone nearly simultaneously posted the indefinite integral version of the problem on mathoverflow: http://mathoverflow.net/questions/105555/integrating-1-xsin1-x-closed2012-08-26

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