I have the following improper integral: $$ \int \limits_{1}^{\infty}\left(\cos\left(\frac{\sin(x)}{\sqrt{x}}\right) - \cos\left(\frac{\cos(x)}{\sqrt{x}}\right)\right)dx $$ And I need to figure out, whether this integral converges absolutely, conditionally or diverges.
I think that it converges conditionally. To show this I did this:
$$ \int \limits_{1}^{\infty}\left(\cos\left(\frac{\sin(x)}{\sqrt{x}}\right) - \cos\left(\frac{\cos(x)}{\sqrt{x}}\right)\right) = -2\int \limits_{1}^{\infty}\left(\sin\left(\frac{\sin(x) + \cos(x)}{2\sqrt{x}}\right)\sin\left(\frac{\sin(x) - \cos(x)}{2\sqrt{x}}\right)\right) $$
Then I used the used the fact that $\sin(x) \sim x, \text{when } x \to 0$, so the following integral is "equivalent":
$$ -2\int \limits_{1}^{\infty} \frac{\cos^2(x) - \sin^2(x)}{2x}dx = -\int \limits_{1}^{\infty} \frac{\cos(2x)}{2x}dx $$
Now I believe it is possible to show that this integral converges (maybe using Dirichlet's test), and we can do same things for integral with module, and then somehow show divergence of it (or maybe convergence).
Could you tell me if I'm doing it correct, or could you give me some hints on how to solve this.