This work uses a recently developed fluctuation free matrix representation method in approximating the integral of the Taylor expansion remainder term. The basis set used for the matrix representation contains common factors of sine and cosine functions with the same frequencies and the same origin. This provides a new numerical univariate integration method to us such that the approximation quality can be controlled by the number of the expansion terms in the Taylor expansion of the integrand and by the dimension of the subspace over which the matrix representations are built. The number of oscillations in the basis set is also another quality control agent and may help to get better approximants in the case of high oscillations. Due to the limitation of space in this extended abstract results of the implementations are left to the presentation.