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Determine if the following integral converges: $$\int_{-\infty}^{\infty}\frac{\cos(x)}{x^3+4x}dx.$$

So far I've thought about using the comparison test but I'm not sure how to implement it. My first thought would be that $\frac{\cos(x)}{x^2+4x}\leq \frac{1}{x(x^2+4)}$ but I am stuck here. Any help with this would be great. Thank you!

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  • $\begingroup$ The divergence of this integral is due to the singularity at $0$. $\endgroup$ – Mark Viola Apr 14 '15 at 14:38
  • $\begingroup$ Note that the integrand is an odd function. $\endgroup$ – science Apr 14 '15 at 14:46
  • $\begingroup$ The integral is undefined. $\endgroup$ – science Apr 14 '15 at 16:19
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Does $$ \int_0^{\pi/3}\frac{1/2}{x^3+x}\,dx $$ converge?

Since $\cos x\ge 1/2$ for $0\le x\le \pi/3$, what can you say?

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  • $\begingroup$ Well this integral diverges so my integral will diverge by the comparison test. Correct? $\endgroup$ – user23793 Apr 14 '15 at 14:43
  • $\begingroup$ @user23793 Yes. $\endgroup$ – egreg Apr 14 '15 at 14:44

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