Show that [Eleft(e^{alpha U(t)} ight)=e^{alpha^{2} t^{3} / 6}] for any constant (alpha), where (U(t)) is the integrated

Show that

\[E\left(e^{\alpha U(t)}\right)=e^{\alpha^{2} t^{3} / 6}\]

for any constant \(\alpha\), where \(U(t)\) is the integrated standard Brownian motion:

\[U(t)=\int_{0}^{t} S(x) d x, t \geq 0\]