A class of simple expressions of increasing accuracy for the free-energy difference between two states is derived based on numerical thermodynamic integration. The implementation of these formulas requires simulations of the initial and fnal (and possibly a few intermediate) states. They involve higher free-energy derivatives at these states which are related to the moments of the probability distribution of the perturbation. Given a specified number of such derivatives, these integration formulas are optimal in the sense that they are exact to the highest possible order of free-energy perturbation theory. The utility of this approach is illustrated for the hydration free energy of water. This problem provides a quite stringent test because the free energy is a highly nonlinear function of the charge so that even fourth order perturbation theory gives a very poor estimate of the free-energy change. Our results should prove most useful for complex, computationally demanding problems where foe-energy differences arise primarily from changes in the electrostatic interactions (e.g., electron transfer, charging of ions, protonation of amino acids in proteins).