Does $\int_{1}^{\infty}\sin(x\log x)dx $ converge? I'm trying to find out whether $\int_{1}^{\infty}\sin(x\log x)dx $ converges, I know that $\int_{1}^{\infty}\sin(x)dx $ diverges but $\int_{1}^{\infty}\sin(x^2)dx $ converges, more than that, $\int_{1}^{\infty}\sin(x^p)dx $ converges for every $p>0$, so it should be converges in infinity. I'd really love your help with this.
Thanks!

This is a version of the Van der Corput lemma, basically.
Note that it's enough to find some $n$ for which $\int_n^{\infty} \sin(x\log(x))\,dx$ converges. The key facts about $f(x) = x\log(x)$ that allow this are a) $\lim_{x \rightarrow \infty} f'(x) = \infty$ and b) $f''(x) > 0$ for $x$ large enough. Specifically, we write
$$\int_n^{\infty} \sin(f(x))\,dx = \int_n^{\infty} f'(x) {\sin(f(x)) \over f'(x)}\,dx$$
$$= \lim_{N \rightarrow \infty} \int_n^{N} (f'(x) \sin(f(x)){1  \over f'(x)}\,dx$$
Integrating the integral on the right by parts you get
$$\int_n^{N} (f'(x) \sin(f(x)){1  \over f'(x)}\,dx = -{\cos(f(N)) \over f'(N)} + {\cos(f(n)) \over f'(n)} + \int_n^N \cos(f(x)){d \over dx}{1 \over f'(x)}$$
$$= -{\cos(f(N)) \over f'(N)} + {\cos(f(n)) \over f'(n)} - \int_n^N \cos(f(x)) {f''(x) \over (f'(x))^2}$$
As $N$ goes to infinity the first term goes to zero since $f'(x)$ goes to $\infty$ as $x$ goes to $\infty$ and $|\cos(f(N))| \leq 1$. The third term is bounded in absolute value by
$$\int_n^N\bigg|{f''(x) \over (f'(x))^2}\bigg|\,dx$$
Since $f''(x) > 0$ we can just take off the absolute values to get
$$\int_n^N{f''(x) \over (f'(x))^2}\,dx$$
Integrating this becomes
$${1 \over f'(N)} - {1 \over f'(n)}$$
Since $f'(N) \rightarrow \infty$ as $N$ goes to $\infty$ this converges as $N$ goes to infinity. Hence the integral $\int_n^{\infty} \cos(f(x)) {f''(x) \over (f'(x))^2}$ converges absolutely, and thus converges.
Hence we have shown the original integral converges.
