We present a variant of the classical integration by parts to introduce a new type of Taylor series expansion and to present some closed forms for integrals involving Jacobi and Laguerre polynomials, which cannot be directly obtained by usual symbolic computation programs, i.e., only some very specific values can be computed by the mentioned programs. An error analysis is given in the sequel for the introduced expansion.
Keywords: Error bound; Generalized Taylor expansion; Integral remainder; Integration by parts; Jacobi and Laguerre polynomials.