  In this paper, we introduce a modification of the free boundary problem
related to optimal stopping problems for diffusion processes. This modification
allows the application of this PDE method in cases where the usual regularity
assumptions on the coefficients and on the gain function are not satisfied. We
apply this method to the optimal stopping of integral functionals with
exponential discount of the form $E_x\int_0^{\tau}e^{-\lambda s}f(X_s) ds$,
$\lambda\ge0$ for one-dimensional diffusions $X$. We prove a general
verification theorem which justifies the modified version of the free boundary
problem. In the case of no drift and discount, the free boundary problem allows
to give a complete and explicit discussion of the stopping problem.
