A flapper equalizer plunger embodies a unitary coiled spring that is compressed by the flow tube to shift the plunger to equalize pressure across the flapper before the flow tube tries to move the flapper. The hydraulic control system then need only to overcome the force of a flapper pivot spring to swing the flapper 90 degrees to an open position behind the flow tube. The integral spring is produced with a wire EDM technique and can be symmetrical or asymmetrical around its periphery as a way of offsetting off-center impact from the bottom of the flow tube. The plunger can be rotationally locked if it is asymmetrical. The pitch can be constant or variable and the output force greatly exceeds the force provided by an independent coiled spring.