We report the first experimental realization of an all-optical temporal integrator. The integrator is implemented using an all-fiber active (gain-assisted) filter based on superimposed fiber Bragg gratings made in an Er-Yb co-doped optical fiber that behaves like an 'optical capacitor'. Functionality of this device was tested by integrating different optical pulses, with time duration down to 60 ps, and by integration of two consecutive pulses that had different relative phases, separated by up to 1 ns. The potential of the developed device for implementing all-optical computing systems for solving ordinary differential equations was also experimentally tested.