Two-dimensional (2D) semiconductors with high carrier mobilities and sizeable bandgap are desirable for future highspeed and low power mechanically flexible nanoelectronics. In this work, we report encapsulated bottom-gated black phosphorus (BP) field-effect transistors (FETs) on flexible polyimide affording maximum carrier mobility of about 310cm(2)/V.s and current on/off ratio exceeding 10(3). Essential circuits of flexible electronic systems enabled by the device ambipolar functionality, high-mobility and current saturation are demonstrated in this work, including digital inverter, frequency doubler, and analog amplifiers featuring a voltage gain of similar to 8.7, which is the state-of-the-art value for flexible 2D semiconductor based amplifiers. In addition, we demonstrate the single FET based flexible BP amplitude-modulated (AM) demodulator, an active stage in radio receivers.