Quite simple looking Integral

Question

I stumbled upon the following integral with positive constants $a$ and $b$ \begin{equation} \int_{-\infty}^{\infty} \exp\left( x - a\mathrm{e}^{bx} \right) dx. \end{equation} but i wasn't able to calculate it... it somehow reminded me of the Gumbel distribution, but i just failed horribly. I was wondering if someone got a hint for me how to calculate it.

Answer 1

Assuming $a,b>0$, by setting $x=\log t$, then by setting $t=u^{1/b}$ and recalling the definition of the $\Gamma$ function, we get: $$\begin{eqnarray*}I(a,b)=\int_{-\infty}^{+\infty}e^{x-a e^{bx}}\,dx = \int_{0}^{+\infty}\exp\left(-a t^b\right)\,dt=\color{red}{a^{-1/b}\cdot \Gamma\left(1+\frac{1}{b}\right)}.\end{eqnarray*}$$