In this paper, some new integral representations are proved for several weighted hypergeometric functions introduced recently in [J.E. Restrepo, A. Kılıc ¸man, P. Agarwal, O. Altun, Adv.Difference Equ., 2017 (2017), 11 pages].Besides, some new subclasses of weighted hypergeometric functions containing the Djrbashian Cauchy type kernel are introduced.The series representing the considered hypergeometric functions are convergent out of some sets of zero ω-capacity, and these hypergeometric functions have finite boundary values everywhere on |z| = 1, out of zero ω-capacity sets.