# Improper Integral with trigonometric functions

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!

• The divergence of this integral is due to the singularity at $0$. – Mark Viola Apr 14 '15 at 14:38
• Note that the integrand is an odd function. – science Apr 14 '15 at 14:46
• The integral is undefined. – 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?