A simple derivation of the integrals of products of Legendre polynomials with logarithmic weight

We explore integrals of products of Legendre polynomials with a logarithmic weight function. More precisely, for Legendre polynomials $P_m$ and $P_n$ of orders $m$ and $n$, respectively, we provide simple derivations of the integrals $$\underset{0}{\overset{1}{\int }}{P}_n(2x-1)P_m(2x-1)\log (x)dx\mathrm{.}$$