i am having though time trying to find if $\int_2^\infty \frac{\cos(x^2)}{x\ln(x)}dx$ converges.
does it converge or absolutely converging?
i am sorry that i am not writing with the correct symbols as i an not used to it yet.
any help will be greatly appreciated thank you.
Change variables with $u = x^2$ to obtain
$$\int_2^c \frac{\cos x^2}{x \ln x }\, dx = \int_4^{c^2} \frac{\cos u}{2u \ln \sqrt{u} }\, du = \int_4^{c^2} \frac{\cos u}{u \ln u }\, du $$
By the Dirichlet test, the improper integral is at least conditionally convergent.
However, the integral fails to converge absolutely since
$$\int_4^{\infty} \frac{|\cos u|}{u \ln u }\, du \geqslant \sum_{k=2}^\infty\int_{\pi/2 + k \pi}^{3\pi/2 + k \pi}\frac{|\cos u|}{u \ln u }\, du \\ \geqslant \sum_{k=2}^\infty\frac{2}{(3\pi/2 + k\pi) \ln (3\pi/2 + k \pi) } \\ \geqslant \sum_{k=2}^\infty\frac{2}{(2k\pi) \ln (2k \pi) } \\ = \frac{1}{\pi}\sum_{k=2}^\infty\frac{1}{k (\ln k + \ln(2\pi)) }$$