Q: Integral similar to a Laplace transform keeps blowing up. I'm working on a problem that results in the integral:
$$
\int_{x=0}^{x\rightarrow\infty}
x^{-1}e^{-xt}
\left[
v(x)
+
t
\int_{x'=x}^{x'\rightarrow\infty}
v(x')dx'
\right]
dx
$$
where $v(x)$ is a probability density function. The integrand without $x^{-1}$ is also a probability density function.  All of my attempts result in something that looks like $\lim_{t\rightarrow\infty}\ln t$.  Also, the resemblance of the expression to the Laplace transform led me to something that had $t\mathscr L\{x^{-1}\}(t)$ which, as far as I can tell, also blows up.
All of this is frustrating because I believe the expression must converge based on the physics I've derived it from.  Any help is greatly appreciated.  Thank you!
