This work is based on the idea of nesting one or more Taylor decompositions in the remainder term of a Taylor decomposition of a function. This provides us with a better approximation quality to the original function. In addition to this basic idea each side of the Taylor decomposition is integrated and the limits of integrations are arranged in such a way to obtain a universal [0, 1] interval without losing from the generality. Thus a univariate approximate integration technique is formed at the cost of getting multivariance in the remainder term. Moreover the remainder term expressed as an integral permits us to apply Fluctuationlessness theorem to it and obtain better results.