An Integral With ln x in the Denominator: ln 5 − ⅓ ln 2

Grimmett and Welsh's prime-faced die exercise has a two-line proof and a hidden assumption. Here is where the prime does its work, and where the claim breaks.

Four integrals, each worked both ways, and the chain-rule test that tells you when to reach for the larger chunk.

A cyclotomic-style denominator and a ln x term look like a convergence problem but are not. The real work is tracking one parameter shift and one factor of 1/6 through a digamma-to-log-gamma step.