A Legendre polynomial integral

Abstract:

Let $\{ {P_n}(x)\}$ be the usual Legendre polynomials. The following integral is apparently new. \[ \int _0^1{P_n}(2x - 1)\log \frac {1}{x}dx = \frac {{{{( - 1)}^n}}}{{n(n + 1)}}\quad {\text {for}}\;n \geqslant 1.\] It has an application in the construction of Gauss quadrature formulas on (0, 1) with weight function $\log (1/x)$.