In this work, the time-domain integration of broadband terahertz (THz) pulses via a tapered two-wire waveguide (TTWWG) is reported. Such a guiding structure consists of two metallic wires separated by a variable air gap that shrinks down to a subwavelength size as the movement takes from the waveguide input to its output. It is shown that while an input THz pulse propagates toward the subwavelength output gap, it is reshaped into its first-order time integral waveform. In order to prove the TTWWG time integration functionality, the THz pulse is detected directly within the output gap of the waveguide, so as to prevent the outcoupling diffraction from altering the shape of the time-integrated THz transient. Since the time-domain integration is due to the tight geometrical confinement of the THz radiation in a subwavelength gap volume, the TTWWG operational spectral range can easily be tuned by judiciously changing both the output gap size and the tapering angle. The results lead to the physical realization of a broadband, analog THz time integrator device, which is envisioned to serve as a key building block for the implementation of complex and ultrahigh-speed analog signal processing operations in THz communication systems.