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Many inversion problems require solution of a Fredholm integral equation of the form , where is the observable, is an operator, and is the unknown parameter distribution. Examples occur in the areas of radiation and scattering, tomography, and geotomography. We reduce the equation to matrix form and apply a maximum entropy technique based on the first principle of data reduction to obtain a most probable distribution. We illustrate the technique by synthetic data examples of geotomography assuming straight rays, with and without noise. The examples show how sharp anomalies may be identified in grossly underdetermined situations. We outline the algorithm used and describe some computational properties. Our method suggests a way of overcoming the ill-conditioned nature of Fredholm integral equation inversion.