  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 $$\int \limits_0^1 P_n(2x-1) P_m(2x-1)\log (x)dx.$$
