  We derive a criterium for the almost sure finiteness of perpetual integrals
of \LL processes for a class of real functions including all continuous
functions and for general one-dimensional L\'evy processes that drifts to plus
infinity. This generalizes previous work of D\"oring and Kyprianou, who
considered L\'evy processes having a local time, leaving the general case as an
open problem. It turns out, that the criterium in the general situation
simplifies significantly in the situation, where the process has a local time,
but we also demonstrate that in general our criterium can not be reduced. This
answers an open problem posed in \cite{doring}.
