The author gives the full description of the integrals of hydrodynamic type \(I=\int P(u)dx\) for the Hamiltonian systems \(u^ i_ t=v^ i_ j(u)u^ j_ x\) \((i,j=1,...,n)\) which arise in hydrodynamic and average theory. Using the generalized hodograph method (an analogue of the method of inverse problem for averaged equations) he presents an explicit procedure for integrating an arbitrary diagonal Hamiltonian system.