This paper considers an integrated multicast and unicast downlink communication system using the non-orthogonal multiple access (NOMA) scheme assisted by an intelligent reflecting surface (IRS). We aim to maximize the unicast data rate while keeping the multicast data rates above a target level by adjusting the reflecting elements of the IRS. The corresponding formulated problem is a nonconvex quadratically constrained quadratic program (QCQP), which is NP-hard. We propose a global optimal algorithm based on branch and bound (BB) and a low-complexity suboptimal algorithm based on semidefinite relaxation (SDR). The simulation results show that the proposed IRS-NOMA scheme outperforms IRS-assisted orthogonal multiple access (OMA) and IRS-NOMA with random phases. Besides, the suboptimal algorithm achieves a better tradeoff between performance and complexity.