Homogeneous broadband and electrically pumped semiconductor radiation sources emitting in the terahertz regime are highly desirable for various applications, including spectroscopy, chemical sensing, and gas identification. In the frequency range between 1 and 5 THz, unipolar quantum cascade lasers employing electron inter-subband transitions in multiple-quantum-well structures are the most powerful semiconductor light sources. However, these devices are normally characterized by either a narrow emission spectrum due to the narrow gain bandwidth of the inter-subband optical transitions or an inhomogeneous broad terahertz spectrum from lasers with heterogeneous stacks of active regions. Here, we report the demonstration of homogeneous spectral spanning of long-cavity terahertz semiconductor quantum cascade lasers based on a bound-to-continuum and resonant phonon design under radio frequency modulation. At a single drive current, the terahertz spectrum under radio frequency modulation continuously spans 330 GHz (~8% of the central frequency), which is the record for single plasmon waveguide terahertz lasers with a bound-to-continuum design. The homogeneous broadband terahertz sources can be used for spectroscopic applications, i.e., GaAs etalon transmission measurement and ammonia gas identification.