The optics of x-ray concentration by tapered glass capillaries is analyzed in terms of a phase-space construction describing their transmission efficiency. The parameters defining the intensity gain are given in terms of parameters describing the x-ray source used, the capillary taper profile, and glass characteristics. We introduce some key concepts in understanding these devices: the extreme ray and a phase-space description of sources and optics. They are used to develop an analytical formulation for the optimum gain characteristics of generalized tapers for use with synchrotrons and other low-divergence sources. This general solution is solved further for the case of conical taper profile. The predictions of this theory are compared with the results of three-dimensional, ray-tracing simulations of x-ray concentration efficiency for conical and paraboloidal tapers.