real analysis - Is the integral $intop_{0}^{infty}frac{cos x}{sqrt{1+x^{3}}}dx$ absolutely convergent, conditionally convergent or divergent?

I'm trying to solve the next problem: Determine if $\intop_{0}^{\infty}\frac{\cos x}{\sqrt{1+x^{3}}}dx$
is absolutetly convergent, conditionally covergent or diverges.

I think that the integral is abosolutely convergent and I tried to
do this: For all $x\geq0$ is true that
$$\frac{\cos x}{\sqrt{1+x^{3}}}\leq\frac{\mid\cos x\mid}{\sqrt{1+x^{3}}}\leq\frac{1}{\sqrt{1+x^{3}}}\leq\frac{1}{\sqrt{x^{3}}}=\frac{1}{x^{3/2}}.$$

Then, using the fact that $\int_{1}^{\infty}\frac{1}{x^{\alpha}}dx$
is convergent for $\alpha>1$ and the comparison test we can conclude
that the integral $\int_{1}^{\infty}\frac{\cos x}{\sqrt{1+x^{3}}}dx$
is absolutely convergent. Also, since the function $f\left(x\right)=\frac{\mid\cos x\mid}{\sqrt{1+x^{3}}}$
is continuous on $[0,\infty)$ then is Riemann integrable on $\left[0,1\right]$.
Therefore,
$$\int_{0}^{\infty}\frac{\mid\cos x\mid}{\sqrt{1+x^{3}}}dx=\int_{0}^{1}\frac{\mid\cos x\mid}{\sqrt{1+x^{3}}}dx+\int_{1}^{\infty}\frac{\mid\cos x\mid}{\sqrt{1+x^{3}}}dx.$$

And then, $\int_{0}^{\infty}\frac{\mid\cos x\mid}{\sqrt{1+x^{3}}}dx$
is convergent since in the last equality, the two sumands on the right
side are finite. Thus, the integral $\int_{0}^{\infty}\frac{\cos x}{\sqrt{1+x^{3}}}dx$
is absolutely convergent.

I don't know if what I did is right. Could you help me checking or
giving me some suggestion?

Thanks.

Answer

This is correct. As you correctly noted, the absolute value of the integrand is continuous on $\Bbb R^+$; and thus in particular Riemann-integrable on any (bounded) interval $[0,I]\subset\Bbb R$.

Also, because you want to prove absolute convergence, your first inequality should be stated as
$$\left|\frac{\cos x}{\sqrt{1+x^3}}\right|\le\frac{|\cos x|}{\sqrt{1+x^{3}}}.$$