<s>[INST] convergence of a integral using comparation 
analyze the comparison criterion if the integral converges or not
$\int\limits_{2}^{+\infty}\frac{\cos x}{2+e^{x^3}}dx$

attempt
i used the fact that $-\frac{1}{2+e^{x^3}}\le\frac{\cos x}{2+e^{x^3}}\le\frac{1}{2+e^{x^3}}$ to get that
$-\int\limits_{2}^{+\infty}\frac{1}{2+e^{x^3}}dx\le\int\limits_{2}^{+\infty}\frac{\cos x}{2+e^{x^3}}dx\le \int\limits_{2}^{+\infty}\frac{1}{2+e^{x^3}}dx$
then for $x>1$ we have that $e^{x^3}>e^{x}\Rightarrow2+e^{x^3}>e^x$ and then $\frac{1}{2+e^{x^3}}<\frac{1}{e^x}=e^{-x}$
and then
$\int\limits_{2}^{+\infty}\frac{1}{2+e^{x^3}}dx<\int\limits_{2}^{+\infty}e^{-x}dx=-e^{-x}\bigg|_{2}^{+\infty}=e^{-2}$
wich mean that $\int\limits_{2}^{+\infty}\frac{\cos x}{2+e^{x^3}}dx$ converge
did my work are correct? has a better way to conclude this?
 [/INST] We have that $\int_{0}^{y}\cos x\,dx $ is bounded and $\frac{1}{2+e^{x^3}}$ is a positive decreasing function whose limit when $x\to +\infty$ is zero, hence the integral is converging by Dirichlet's criterion, for instance.
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