Does the following integral converge?
$$\int_1^\infty \sin^2 (x^2) \, dx$$
I tried $$\int_1^\infty \sin^2(x^2) \, dx=\int_1^\infty \frac{1-\cos(2x^2)}{2} \, dx = \frac{\sqrt{2}}{4} \int_1^\infty\frac{1-\cos(u)}{2\sqrt{u}} \, du$$
The idea is that, I want to compare the original integral to a divergent $p$-integral. But I am not sure how to proceed from here.