Unmanned aerial vehicle (UAV) has high flexibility and controllable mobility, therefore it is considered as a promising enabler for future integrated sensing and communication (ISAC). In this paper, we propose a novel adaptable ISAC (AISAC) mechanism in the UAV-enabled system, where the UAV performs sensing on demand during communication and the sensing duration is configured flexibly according to the application requirements rather than keeping the same with the communication duration. Our designed mechanism avoids the excessive sensing and waste of radio resources, therefore improving the resource utilization and system performance. In the UAV-enabled AISAC system, we aim at maximizing the average system throughput by optimizing the communication and sensing beamforming as well as UAV trajectory while guaranteeing the quality-of-service requirements of communication and sensing. To efficiently solve the considered non-convex optimization problem, we first propose an efficient alternating optimization algorithm to optimize the communication and sensing beamforming for a given UAV location, and then develop a low-complexity joint beamforming and UAV trajectory optimization algorithm that sequentially searches the optimal UAV location until reaching the final location. Numerical results validate the superiority of the proposed adaptable mechanism and the effectiveness of the designed algorithm.