An extension of trapezoidal type product integration rules (Q843172)

Let \(-\infty < a < b< \infty\) and let \(F_0 \in C[a,b]\) be a positive weight function. The author presents a generalized Euler-Maclaurin summation formula for an integral \[ \int_a^b F_0(x) \,g(x)\,{\mathrm d}x\,, \] where \(g\in C^2[a,b]\). This method can be seen as a generalization of the trapezoidal rule. The order of the corresponding error is estimated. Numerical examples are given.