This work concerns with the Taylor expansion remainder term evaluation. The integral defining the remainder is standardized in interval first, by using an appropriate affine transformation. Then it is approximated by the utilization of the fluctuation free integration which was developed quite recently. This method approximates the matrix representation of the function to be integrated in terms of the matrix representation of the independent variable. The approximation quality depends on a lot of issues like curvature, smoothness, singularities, and the basis function used in the representation. This work investigates the basis functions containing a common factor of exponential function times an appropriate power of the independent variable. The purpose is to investigate the role of the decaying nature existing in the basis functions on the approximation quality.