Power rule for integration ($\int x^{q}dx=\frac{x^{q+1}}{q+1}+C,q\ne -1$)

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 $\log x+C$.