I want to know about the convergence of the following integral :
$$ \int_1^{+\infty}\sin(x^3+x)dx $$
I have tried a few things here and there but I am nowhere close to prove the convergence or divergence.
I was also thinking about the following way :
If $I\subset[1,\infty)$ is any compact interval, then, if possible, we can show that $\int_{I}\sin(x^3+x)dx$ is uniformly bounded.
But I am still unable to prove this.
I would be grateful for any hint or help.
HINT
$$\int_1^{+\infty}\sin(x^3+x)dx=\int_1^{+\infty}\frac{1}{3x^2+1}\cdot(3x^2+1)\sin(x^3+x)dx$$
Now, integrate by parts with $dv=(3x^2+1)\sin(x^3+x)$
Can you finish?