Electro-optic phase modulators are critical components in modern communication, microwave photonic, and quantum photonic systems. Important for these applications is to achieve modulators with low half-wave voltage at high frequencies. Here we demonstrate an integrated phase modulator, based on a thin-film lithium niobate platform, that simultaneously features small on-chip loss (~ 1 dB) and low half-wave voltage over a large spectral range (3.5 - 4.5 V at 5 - 40 GHz). By driving the modulator with a strong 30-GHz microwave signal corresponding to around four half-wave voltages, we generate an optical frequency comb consisting of over 40 sidebands spanning 10 nm in the telecom L-band. The high electro-optic performance combined with the high RF power-handling ability (3.1 W) of our integrated phase modulator are crucial for future photonics and microwave systems.