Inexact analogy with the electronic counterparts, a photonic integrator is a fundamental component for constructing all-optical circuits used in information processing and computing. In this paper, we propose and experimentally demonstrate an all-optical microwave envelope integrator for executing periodic signals, based on two cascaded 2-tap microwave photonic filters (MPFs). By properly arranging the weights of the taps and the unit delays of the two MPFs, the frequency response of the two cascaded 2-tap MPFs can closely approach that of an ideal integrator. Consequently, the integrated output of the input signal is obtained. Theoretical analysis and simulations are given in detail. Experimentally, when a square waveform, a sawtooth waveform, or a reversed-sawtooth waveform is used as input signal, the corresponding integrated output, i.e., a triangular waveform, a bright parabola, or a dark parabola, is successfully obtained, which agree well with the theoretical expectations. The proposed scheme exhibits not only anew way to implement an all-optical microwave envelope integrator, but also the advantages of large operation bandwidth, simple configuration, easy operation, and low cost. This scheme shows great potential for all-optical signal processing applications.