For my purposes, $n$ is a non-negative integer, and $x > 0$. I didn't know how to evaluate this integral, so I plugged it into Mathematica. It told me the solution is
$(-1)^n B(e^x; -n, n+1)$
I have never before seen the incomplete Beta function defined for negative integers, though I did find a reference with a definition over negative integers here.
Confusingly, asking Mathematica to evaluate this solution for a particular value of $n$ always gives ComplexInfinity
. Does this mean Mathematica doesn't know a definition for the incomplete Beta function with negative integers? And if not, why would it provide me with such a solution?
Due to the conflicting results from Mathematica, I am not confident this solution is actually correct/complete. Can somebody point me towards a more detailed solution?