Integral of Power

Theorem
$\displaystyle \forall n \in \R, n \ne -1: \int_0^b x^n \mathrm d x = \frac {b^{n+1}} {n+1}$

Fermat's Proof
This proof is is valid only for positive rational numbers, that is, it proves that:
 * $\displaystyle \forall n \in \Q_{>0} : \int_0^b x^n \mathrm d x = \frac {b^{n+1}} {n+1}$

Comment
The conventional proof of course holds for all real $n \ne -1$, not just where $n$ is a strictly positive rational.

However, the real point of this page is Fermat's proof, which demonstrates how integration was achieved before the full machinery of calculus had been thoroughly constructed.

Cavalieri had previously made progress with this problem, proving it for integral $1 \le n \le 9$ but the algebra for the proof of each power was more difficult than the previous one, and he found $10$ too much hard work. The clear beauty of Fermat's approach was that it works for all $n$, rational as well as integral.