A family of definite integrals involving Legendre’s polynomials

The main objective of this article is to provide the analytical solutions (not previously found and not available in the literature) of some problems related with definite integrals integrands of which are the products of the derivatives of Legendre’s polynomials of first kind having different order, with the help of some derivatives of Legendre’s polynomials of first kind P_n(x), Rodrigues formula, Leibnitz’s generalized rule for successive integration by parts and certain values of successive differential coefficients of (x^2 − 1)^r at x = ±1.