Referring to trivial form, sometimes called Cavalieri's formula: $$ \int x^p \mathrm{d}x=\frac{x^{p+1}}{p+1} + C, \qquad p\neq-1 $$ I was wondering what other restrictions on $p$ exist besides $p \neq -1$.
Initially I thought $p \in \mathbb{Z}\backslash\{-1\}$, but the formula is also used to obtain forms for radicals and root functions. Then I thought perhaps $p \in \mathbb{R}\backslash\{-1\}$, but then I doubted: would it work on complex numbers? I carried out some integrations with $p\in \mathbb{C}\backslash\{-1\}$, and the formula seems to work fine.
I could not find a proof with any restrictions on $p$. Any ideas on how general can $p$ become without breaking the equality?
Thank you for the insight.