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!

  • $\begingroup$ 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

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?

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.