Distributed event-triggered consensus tracking is researched for multi-agent systems with general linear dynamics under directed switching topologies. Most existing works focus on event-triggered control with fixed communication topology, which however can not meet the demand that the relations among agents may vary or the communication topology may change over time. In light of this, an integral-based event-triggered control strategy is developed in the consideration of switching topologies. Firstly, when the communication topology contains a directed spanning tree with the leader being the root, we solve the integral-type event-triggered leader-following consensus under switching topologies by using Lyapunov stability theory, algebraic graph theory and matrix transformation. Secondly, integral-type event-triggered containment control of multi-agent systems is investigated under switching topologies when there exist multiple leaders. Moreover, the Zeno-behavior of the closed-loop systems is excluded by developing a positive constant lower bound under the proposed control rules in both above-mentioned cases. Finally, simulations are presented to verify the validity of the proposed control strategy.