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!
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?