  Let $X$ be a linear diffusion and $f$ a non-negative, Borel measurable
function. We are interested in finding conditions on $X$ and $f$ which imply
that the perpetual integral functional $$ I^X_\infty(f):=\int_0^\infty f(X_t)
dt $$ is identical in law with the first hitting time of a point for some other
diffusion. This phenomenon may often be explained using random time change.
Because of some potential applications in mathematical finance, we are
considering mainly the case when $X$ is a Brownian motion with drift $\mu>0,$
denoted $\{B^{(\mu)}_t: t\geq 0\},$ but it is obvious that the method presented
is more general. We also review the known examples and give new ones. In
particular, results concerning one-sided functionals $$ \int_0^\infty
f(B^{(\mu)}_t) {\bf 1}_{\{B^{(\mu)}_t<0\}} dt\quad {\rm and}\quad \int_0^\infty
f(B^{(\mu)}_t) {\bf 1}_{\{B^{(\mu)}_t>0\}} dt $$ are presented.
  This approach generalizes the proof, based on the random time change
techniques, of the fact that the Dufresne functional (this corresponds to
$f(x)=\exp(-2x)),$ playing quite an important r\^ole in the study of geometric
Brownian motion, is identical in law with the first hitting time for a Bessel
process. Another functional arising naturally in this context is %associated to
the function $$ \int_0^\infty \big(a+\exp(B^{(\mu)}_t)\big)^{-2} dt, $$ which
is seen, in the case $\mu=1/2,$ to be identical in law with the first hitting
time for a Brownian motion with drift $\mu=a/2.$
  The paper is concluded by discussing how the Feynman-Kac formula can be used
to find the distribution of a perpetual integral functional.
