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We present a new high-order integral algorithm for the solution of scattering problems by heterogeneous bodies under TE radiation. Here, a scatterer is represented by a (continuously or discontinuously) varying refractive index n(x) within a two-dimensional (2-D) bounded region. Solutions of the associated Helmholtz equation under given incident fields are then obtained by high-order inversion of the Lippmann-Schwinger integral equation. The algorithm runs in O(N log(N)) operations, where N is the number of discretization points. Our method provides highly accurate solutions in short computing times, even for problems in which the scattering bodies contain complex geometric singularities.