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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?

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    $\begingroup$ $\int(1-e^{-x})^n\,dx=x-\sum_{p=1}^n\tfrac{1}{p}(1-e^{-x})^p$ $\endgroup$ Mar 27, 2015 at 19:26

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