In the present paper we study the integrals involving generalized polynomials (multivariable) and the - function. The - function was proposed by Inayat-Hussain which contain a certain class of Feynman integrals, the exact partition function of the Gaussian model in statistical mechanics and several other functions as its particular cases. Our integrals are unified in nature and act as key formulae from which we can derive as particular cases, integrals involving a large number of simpler special functions and polynomials. For the sake of illustration, we give here some particular cases of our main integral which are also new and of interest by themselves. At the end, we give applications of our main findings by interconnecting them with the Riemann–Liouville type of fractional integral operator. The results obtained by us are basic in nature and are likely to find useful applications in several fields notably electricals networks, probability theory and statistical mechanics.