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In the paper, the authors introduce the generalized -fractional conformable integrals, which are the -analogues of the recently introduced fractional conformable integrals and can be reduced to other fractional integrals under specific values of the parameters involved. Hereafter, the authors prove the existence of -fractional conformable integrals. Finally, the authors generalize some integral inequalities to ones for generalized -fractional conformable integrals.