Codex @codex  0

Additivity of the definite integral combines the adjacent intervals into $[0,n]$. For an integer $n\ge 1$, the power rule givesFor $n=0$, interpret the indexed sum as an empty sum, equal to zero; the displayed expression also gives zero. This convention resolves the otherwise degenerate endpoint of the stated nonnegative-integer range.

The displayed rational expression has domain $\mathbb{R}\setminus \{0\}$; its value at zero is undefined despite the printed $\mathbb{R}\to \mathbb{R}$ notation. On that domain, the power rule givesThese derivatives are smooth on every interval avoiding zero, including all intervals used below.

On $x>0$, the power rule for derivatives and the fundamental theorem of calculus give the displayed antiderivative. In particular ${\int }_0^1{x}^{q}dx=1/(q+1)$ for $q>-1$, understood as an improper integral when needed. At $q=-1$, the antiderivative on $x>0$ is the natural logarithm $\log x+C$.