Recently developed fluctuation free integration is beyond its particular cases, the Gauss quadratures, where the basis set is composed of polynomials. It is mainly based on the matrix representations of the function operators and the efficiency of the method shows a broad band of levels from poor to excellent. The selection of the basis function set on which the matrix representation is based plays a key role on this efficiency and the method works well for smooth integrands. If the function to be integrated is highly oscillatory efficiency drastically drops. However, it is possible to regain the efficiency by using high frequency asymptotic expansions and separating the high oscillatory parts out to the weight of the integration. This work is just conceptual and aims at the construction of a high efficient numerical approximation method.