A detailed presentation of the recently introduced integration-free method, with applications to determine the energy levels of the generalised quantum anharmonic oscillators, are given. Numerical calculations are realised for the quartic and the sextic oscillators. Energy eigenvalues obtained for the ground state as well as for the first few excited states accurate to thirty digits are very impressive and demonstrate the efficiency of the method. Certain remarks about the selection of the

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... functions and a convergence discussion on the presented simple approximation scheme are also included in this paper.