Summary: This paper presents a simple and efficient method for the design of recursive digital fractional order integrator when the order of integration is a real number between 0 and 1. The proposed method is based on the impulse invariance method. First the initial value theorem is used for the selection of the initial value of the impulse response and then any of the well-established signal modeling techniques can be employed for the parameterization of the discrete impulse response by pole-zero models. For a given model order, the approximation accuracy greatly depends on the initial value selected. Illustrative examples are presented to demonstrate the effectiveness and usefulness of the proposed method.