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The aim of the present paper is to evaluate new integral relations involving a general class of polynomials and I-functions. The values of the relations are obtained in terms of the Psi functions y (z) (the logarithmic derivative of Γ(z)). These integral relations are unified in nature and act as a key formula from which we can obtain their special cases. For the sake of illustration, we record here some special cases of our main formulas which are also new and known. The formulas established here are basic in nature and are likely to find useful applications in the field of science and engineering. doi:10.14456/WJST.2015.38