Solution
= Solution
Near the endpoint $t=0$,
$$
\frac1{1+t}\sim\sum_{n=0}^\infty(-1)^nt^n.
$$
The <Watson lemma> integrates this local expansion term by term, and
$$
\int_0^\infty e^{-xt}t^n\,dt
=\frac{\Gamma(n+1)}{x^{n+1}}
=\frac{n!}{x^{n+1}}.
$$
Therefore the <reciprocal-amplitude Laplace integral> has expansion
$$
\boxed{
U(x)\sim\sum_{n=0}^\infty
(-1)^nn!x^{-n-1}},
\qquad x\to+\infty,
$$
so
$$
\boxed{a_n=(-1)^nn!}.
$$