The power-function integration formula for arbitrary real exponents

Let $a,b\in\mathbb R$ and let $p\in\mathbb R$ be an exponent for which the integral and the displayed powers are defined. Power-function integration conjecture. The integration formula

$$\int_a^b x^p\,dx=\frac{b^{p+1}-a^{p+1}}{p+1}$$

should hold for every real exponent $p\in\mathbb R$. The formula requires qualification at $p=-1$ and when $x^p$ is not real-valued on the interval, so the statement as written is not valid without additional hypotheses; its status should therefore be checked against the intended domain and treatment of the exceptional case.