  We consider expectation values of local operators in (continuum) integrable
models in a situation when the mean value is calculated in a single Bethe state
with a large number of particles. We develop a form factor expansion for the
thermodynamic limit of the mean value, which applies whenever the distribution
of Bethe roots is given by smooth density functions. We present three
applications of our general result: i) In the framework of integrable Quantum
Field Theory (IQFT) we present a derivation of the LeClair-Mussardo formula for
finite temperature one-point functions. We also extend the results to boundary
operators in Boundary Field Theories. ii) We establish the LeClair-Mussardo
formula for the non-relativistic 1D Bose gas in the framework of Algebraic
Bethe Ansatz (ABA). This way we obtain an alternative derivation of the results
of Kormos et. al. for the (temperature dependent) local correlations using only
the concepts of ABA. iii) In IQFT we consider the long-time limit of one-point
functions after a certain type of global quench. It is shown that our general
results imply the integral series found by Fioretti and Mussardo. We also
discuss the generalized Eigenstate Thermalization hypothesis in the context of
quantum quenches in integrable models. It is shown that a single mean value
always takes the form of a thermodynamic average in a Generalized Gibbs
Ensemble, although the relation to the conserved charges is rather indirect.
