An Extended Frullani Integral

Question

In the development of Methodology $2$ of This Answer, I found a possible new extension of Frullani's Integral (See Here).

Theorem: Let $f$ be Riemann integrable on $[0,x]$ for all $x>0$ and let $a>0$ and $b>0$. Furthermore, let $F(x)=\int_0^x f(t)\,dt$ denote an antiderivative of $f(t)$ and $\bar F(x)=\frac1xF(x)$ be the average value of $f$ on the interval $[0,x]$.

If the integral $ \int_0^\infty \frac{f(ax)-f(bx)}{x}\,dx$ exists and $\lim_{x\to\infty}\bar F(x)=\bar F_\infty$ and $\lim_{x\to0^+}\bar F(x)= F'(0)$ exist and are finite, then

$$\int_0^\infty \frac{f(ax)-f(bx)}{x}\,dx=(F'(0)-\bar F_\infty)\log(b/a)\tag1$$

Proof:

Integrating by parts the integral on the left-hand side of $(1)$ with $u=\frac1x$ and $v=\frac1aF(ax)-\frac1bF(bx)$, we obtain

$$\int_0^\infty \frac{f(ax)-f(bx)}{x}\,dx=\int_0^\infty \frac{\bar F(ax)-\bar F(bx)}{x}\,dx\tag2$$

Note that $(2)$ is a standard Frullani integral and the result in $(1)$ follows immediately.

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Example Applications:

Alongside the integral in This Answer, namley $\int_0^\infty \frac{|\sin(\sqrt{qx})|-|\sin(\sqrt{px})|}{x}\,dx=\frac2\pi \log(q/p)$, we can use $(1)$ to evaluate the companion integral

$$\int_0^\infty \frac{|\cos(\sqrt{qx})|-|\cos(\sqrt{px})|}{x}\,dx=\left(1-\frac2\pi\right)\log(p/q)$$

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QUESTIONS:

Is there a reference to the Theorem herein or is this a new result? If so, please advise?

Alongside the two examples I cited, are there other non-standard Frullani integrals to which this result would not only apply, but facilitate evaluation?

Can the conditions of the theorem be relaxed?

Can the result be generalized further? For example, in This Answer, I generalized Frullani for complex $a$ and $b$, but I required $f$ to be analytic.