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I am trying to determine whether $$ \int_{-3}^{2} \frac{xdx}{x^3+2x+2}\,.$$ converges or diverges

Usually, for improper integrals where there is a vertical asymptote, I try to split the integral up around the asymptote, but I can't really do that here cause I cannot factor it. Any recommendations?

Thanks

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  • $\begingroup$ This integral does not converge on the given interval $\endgroup$ – Dr. Sonnhard Graubner Nov 4 '18 at 18:34
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    $\begingroup$ How would I actually go about proving that? I thought the same thing when I looked at the graph. $\endgroup$ – Flyrom Nov 4 '18 at 18:39
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Hint: The denominator is positive at $0$ and negative at $-1,$ hence must be $0$ somewhere in between.

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  • $\begingroup$ Yeah I realized that when I saw the graph. Should I just approximate the asymptote and do comparison tests on the approximate sub-intervals? $\endgroup$ – Flyrom Nov 4 '18 at 18:41

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