I am new here, this is the first post I make. I am statistician who is rusty in calculus. Also I know I must check how to write in some kind of math (or latex) environment in this forum and any hints will be welcome. I should check the site some more of course.

So, let following integral

int ( exp(tx-g (lx)^a ) * x^(ag-1) ,from 0 to Inf) dx

where -Inf<a<Inf, l>0, g>0, -Inf<t<Inf.

I don't want to solve it, I just want to know when does it exist (and I think it is not easy).

I derived that it exists for all t<0. I think that the whole "game" will take place with the possible values of "a" and "t". There might be some problems if a>0 and t>0? or if 0<a<1 maybe.

I am a little confused and tried some possible values in the computer to see when it does exists but I haven't came up with any general constraints.

Thanx in advance for any help!