Let B(µ) denote a Brownian motion with drift µ. In this paper we study two perpetual integral functionals of B(µ). The first one, introduced and investigated by Dufresne in [5], is
It is known that this functional is identical in law with the first hitting time of 0 for a Bessel process with index µ. In particular, we analyze the following reflected (or one-sided) variants of Dufresne's functional
and
We shall show in this paper how these functionals can also be connected to hitting times. Our second functional, which we call Dufresne's translated functional, is
where c and ν are positive. This functional has all its moments finite, in contrast to Dufresne's functional which has only some finite moments. We compute explicitly the Laplace transform of 
in the case ν = 1/2 (other parameter values do not seem to allow explicit solutions) and connect this variable, as well as its reflected variants, to hitting times.
