# $\int_{0}^{\infty}\sin({x f({x})}) d x$ converges for positive monotone increasing unbounded continuous $f({x})$

Let a continuous speed function $$f:\mathbb{R}^+\to\mathbb{R}^+$$ exist such that $$[{\{a,b\}\subseteq\mathbb{R}^+\land a. Then $$\int_{0}^{\infty}\sin({x f({x})}) d x$$ converges.

Looking at the positive and negative parts in pairs, if the integral starts with a positive chunk, the integral is positive, and if the integral starts with a negative chunk, then the integral is negative. Thus the first chunk is an upper bound if the chunk is positive, and the first chunk is a lower bound if the chunk is negative. The chunks approach zero because $$f(x)\to\infty$$ as $$x\to\infty$$, thus the upper and lower bounds approach each other.

My problem with this argument is that even though this argument is obvious, the argument seems non-rigorous.

I need the proof to convince Wolfram staff about a bug report: Integrate[Sin[x*Log[x+1]], {x, 0, Infinity}] is claimed to not converge because it converges too slowly.