How do I approach showing that integral $$\int^{\infty}_0\sin^3(x^2+2x)\,\mathrm dx$$ converges absolutely/conditionally? I have calculated via software that $\int^{\infty}_0\sin^3(x^2+2x)\,\mathrm dx$ is indeed convergent, but how is it done exactly?
What I tried:
Let $U = x^2+2x, N\in \mathbb{N}, N>1$, then
$$\int^{2\pi N}_0|\sin^3(u)|\,\mathrm dx=\sum^{N-1}_{n=0}\int^{2\pi (n+1)}_{2\pi n}|\sin^3(u)|\,\mathrm du\leq \cdots$$
I am not sure what I even did is correct. I would appreciate a hint :)