This work aims at the evaluation of a univariate integral whose integrand except the weight function is assumed to be analytic inside a region containing the integration interval as an interior line in the complex plane of the independent variable. The method developed here is based on the idea of fluctuation removal which is the basic root of fluctuation free integration and Gauss quadratures. To this end, first, weight function generating subspaces are constructed. Then various possibilities to use a true weight function in the integration are discussed. All analyses here are for one node integration although higher order fluctuation removal possibilities are also restrictedly discussed. Certain illustrative applications are also distributed in side the sections to support the ideas of the work.