If anyone can help me solve the following equation I will really appreciate it. It is not part of any assignment or DIY kind of thing. I am trying to solve one research paper and it is part of the bigger problem. Seriously stuck here for two days.
$\int^\infty_0 t(n\lambda e^{-ne^{-\lambda t} - \lambda t}) dt$
Appreciate
Hint. By making the change of variable $$ u=e^{-\lambda t},\, (\lambda>0)\qquad du= -\lambda e^{-\lambda t}dt, \qquad -\lambda t=\ln u, $$ you obtain $$ \int^\infty_0 t(n\lambda e^{-ne^{-\lambda t} - \lambda t})dt=-\frac{n}{\lambda}\int_0^1\ln u \:e^{-nu}du $$ then by the change of variable $v=nu$ you get an answer in terms of the Euler constant and in terms of the incomplete gamma function.