Let a continuous speed function $f:\mathbb{R}^+\to\mathbb{R}^+$ exist such that $ [{\{a,b\}\subseteq\mathbb{R}^+\land a<b}] \Rightarrow f(a)<f(b) $. 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.


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