Why does $\int_1^\infty \sin(x \log x)\,\mathrm{d}x$ converge?

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  • Does the integral $\int_{1}^{\infty} \sin(x\log x) \,\mathrm{d}x$ converge?

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Let’s take a look at a graph.

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It appears that the graph oscillates up and down over the $x$-axis and that the labeled areas are decreasing to zero from left to right. In fact, it’s not hard to prove these statements, namely

$$\int_1^\infty \sin(x\log x)\,dx = \sum_{n=1}^{\infty} (-1)^{n+1} I_n$$

and that $\lim_{n\rightarrow\infty} I_n = 0$ with $0<I_{n+1}<I_n$. This boils down to the fact the area $I_k$ is bound on the left by the $k^\text{th}$ of $sin(x\log(x))$ and on the right by the next root. Furthermore, these roots are exactly the spots where
$$x\log(x) = k\pi,$$
for some integer $k$. The fact that the distance between consecutive roots tends to zero boils down to the fact that $x\log(x)$ is an increasing, convex function.

enter image description here