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 Pₘ and Pₙ of orders m and n, respectively, we provide simple derivations of the integrals ₀¹ Pₙ (2x-1) Pₘ (2x-1) (x) dx.