Physicist here, so forgive me if I'm being a bit sloppy.
I was considering the integrals $$ \tau(s) = \int_{0}^{L}\frac{{\rm d}x}{\sqrt{1-f(x)/s}} $$ for all $s>\max\{f\}$, and I came to wonder about the inverse problem. How much can be said about $f(x)$ if we were only given $\tau(s)$?
The extra conditions that I have are that $f\ge0$ for all $x\in[0,L]$ and $$ \int_{0}^{L}f(x)\,{\rm d}x = F $$ is finite.
Alternatively, I could see that the integration limits could be expanded to infinity and redefining the transform as $$ \tau(s) = \int_{-\infty}^{\infty}\left[\frac{1}{\sqrt{1-f(x)/s}}-1\right]{\rm d}x =\int_{-\infty}^{\infty}\frac{1-\sqrt{1-f(x)/s}}{\sqrt{1-f(x)/s}}{\rm\,d}x. $$