On the basis of fractional calculus, Y. Hu and D. Nualart (2009) introduced an alternative approach to the fundamental theory of rough path analysis. A purpose of the speaker's study is to develop the approach by Hu and Nualart (2009). In this talk, using fractional calculus, we will introduce an approach to the rough path integral introduced by M. Gubinelli (2004). Our definition of the integral is given explicitly by the Lebesgue integrals for fractional derivatives. We will show that our definition of the integral is consistent with the usual definition, given by the limit of the compensated Riemann--Stieltjes sums. We will also explain that this result provides such an explicit expression of the rough path integral introduced by T. J. Lyons (1998). Our result is a generalization of that of Hu and Nualart (2009), and our proof is based on a method by M. Z\"ahle (1998).