In a gradient index (GRIN) material, the index profile, in general, is given by a polynomial. The curved light rays are computed by a step-by-step integration method. Sharma et al., have given the details of the Runge-Kutta method; it is time-consuming and approximate. GRIN materials have now started being offered on the market place. When the gradient is axial (i.e., when the index varies with the depth in the lens, along the optical axis) it is the most useful for optical design of real-life large-aperture imaging systems. These axial gradients, that are offered for sale, are almost linear. Moore et al., have explained their efforts to obtain more linear profiles. When the gradient is exactly linear, the differential equation can be exactly integrated, which saves time for preliminary design through reduced computing time. This paper explains the details of this integration, gives the exact parametric equation of a light ray, and, hence, derives paraxial formulas for focal lengths.