To explore the use of intelligent reflecting surface (IRS) technology to improve target detection in an integrated sensing and communication (ISAC) system, this paper investigates the joint active and passive beamforming design for ISAC system equipped with IRS. An optimization problem is formulated to maximize the signal-to-noise ratio (SNR) at the base station (BS) while ensuring a minimum signal-to-interference-plus-noise-ratio (SINR) at each communication user (CU). An alternative algorithm is proposed to tackle this non-convex problem, and the problem is decomposed into three sub-problems. In the first sub-problem, the semi-definite relaxation (SDR) algorithm is used to solve the communication and sensing beamformers. A receive combining vector at the base station is derived from an equivalent Rayleigh-quotient problem in the second sub-problem. Lastly, the Successive Convex Approximation (SCA) based algorithm is applied to yield the IRS phase shift solution in the third sub-problem. The optimization algorithm alternates between these three steps until convergence is achieved. Simulation results demonstrate the effectiveness of the proposed beamforming algorithm, showcasing its superiority over the matched filter (MF) approach.