TITLE: Hints on integrating (rather complicated) exponential function
QUESTION [7 upvotes]: I am interested in computing the following integral
$$\int_0^\infty \frac{1}{\sqrt{t}}\exp\left(-\frac{(x-t)^2}{t} - t\right)\, dt. $$
A hint is good enough for me!

REPLY [6 votes]: Hint. Make the change of variable
$$
u=\sqrt{t}, \quad du=\frac{dt}{2\sqrt{t}},
$$ giving
$$
\int_0^\infty \frac{1}{\sqrt{t}}\exp\left(-\frac{(x-t)^2}{t} - t\right)\, dt=\int_{-\infty}^\infty \exp\left(-2u^2-\frac{x^2}{u^2}-2 x\right)\, du
$$ then use G. Boole's result:
$$
\int_{-\infty}^{+\infty}f\left(u-\frac{a}{u}\right)\mathrm{d}u=\int_{-\infty}^{+\infty} f(u)\: \mathrm{d}u,\quad a>0.
$$