The free-energy change in a system represented by an empirical energy function, when calculated :by thermodynamic integration from the initial to the final state, can be expressed as a sum of "free-energy components," that correspond to contributions from different energy terms or different parts of the system. Although the overall free energy is path independent, the values of the components depend on the integration path, which determines the physical processes involved in the change. A cumulant expansion is used to express the total free-energy change as a sum of terms that contain the cumulants of a single energy term plus a coupling term involving cumulants of more than one energy term. The dependence of the free-energy components on the thermodynamic integration path is shown to arise from the different partitionings of the latter term. A concerted linear integration path, along which the total energy function changes from the initial to the final state as a linear function of an external parameter lambda, as well as piecewise-linear paths, along which individual energy terms are changed linearly, are examined. Along the concerted linear path the partitioning of the coupling term preserves permutation symmetry with respect to energy components to all orders. This contrasts to piecewise-linear paths, where the total coupling is projected onto one component. To illustrate the analysis, calculations are made for the transformation. Cl-(0)-->Br-(-) (a neutral particle with the van der Waals parameters of Cl-(-) to a charged particle with the van der Waals parameters of Br(-)) in aqueous solution along various paths. Terms through third order in the cumulant expansion for the change in the solvation free energy are evaluated by use of the reference interaction site model with the hypernetted chain closure, and the values obtained along the linear and piecewise-linear paths are compared and analyzed. The utility of cumulants for the prediction of the effect of specific changes in the energy terms, and for interpreting free-energy components is discussed. In most cases a first-order approximation is not accurate enough to be useful, but results through third order are accurate to 5% or better. In the Appendix a general expression for the free-energy derivatives of a Hamiltonian with a general functional dependence on a parameter lambda is derived.