2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66561We prove that $$\int_0^1\frac{-\log f(x)}xdx=\frac{π^2}{3ab}$$ where $f(x)$ is the decreasing function that satisfies $f^a-f^b=x^a-x^b$, for $0<a<b$. When $a$ is an integer and $b=a+1$ we deduce several combinatorial results. These include an asymptotic formula for the number of integer partitions not having $a$ consecutive parts, and a formula for the metastability thresholds of a class of threshold growth cellular automaton models related to bootstrap percolation.Revised version. 28 pages, 2 figuresProbabilityCombinatoricsNumber Theory26A06; 05A17; 60C05; 60K35text