In this work we come up with the statement of the Fluctuationlessness theorem recently conjectured and proven by M. Demiralp and its application to numerical integration of univariate functions by restructuring the Taylor expansion with explicit remainder term. The Fluctuationlessness theorem is stated. Following this step an orthonormal basis set is formed and the necessary formulae for calculating the coefficients of the three term recursion formula are constructed. Then for multivariate numerical integration, instead of dealing with a single formula for multiple remainder terms, a new approach that is already mentioned for bivariate functions is taken into consideration. At every step of a multivariate integration one variable is considered and the others are held constant. In such a way, this gives us the possibility to get rid of the complexity of calculations. The trivariate case is taken into account and its generalization is step by step explained. At the final stage implementations are done for some trivariate functions and the results are tabulated together with the implementation times.