Moving Gumbel cutoff in an inverse-power tail (${\int }_a^{\infty }{x}^{-r}{e}^{-\lambda e^{-x}}dx\sim \frac{L^{1-r}}{r-1}-\gamma L^{-r}(L=\log \lambda )$)

For fixed $a>0$ and $r>1$, the factor $\exp (-\lambda e^{-x})$ is a translated Gumbel distribution cumulative function, with transition near $x=L$. Integration by parts transforms the integral to an exponentially small boundary term plus $(r-1{)}^{-1}{\int }_0^{\lambda e^{-a}}{e}^{-u}(L-\log u{)}^{1-r}du$. Expanding the latter against its finite logarithmic moments gives the displayed two terms because ${\int }_0^{\infty }{e}^{-u}\log udu=-\gamma$, with $\gamma$ Euler's constant. The smooth cutoff shifts the first correction even though replacing it by a step gives the leading tail.