In operator theory, there is an important problem called the invariant subspace problem. This important problem of mathematics has been clear for more than half a century. However, the solution seems to be nowhere in sight. With this motivation, we investigate the invariant subspaces of the fractional integral operator in the Banach space with certain conditions in this paper. Also, by using the Duhamel product method, unicellularity of the fractional integral operator on some space is obtained and the description of the invariant subspaces is given.