Prove that:

$\displaystyle \int_{a}^{b} x^n\cdot f^{n + 1}(x) dx = F(b) - F(a)$

where,

● $\displaystyle F(x) = x^nf^n(x) = nx^{n - 1}f^{n - 1}(x) + n(n - 1)x^{n - 2}f^{n - 2}(x) - ... + (-1)^n n!f(x)$

● $\displaystyle f^r(x) = \frac{d^rf(x)}{dx^r}$ for $\displaystyle 0< r\leq n$ ; $\displaystyle r,n\in\mathbb{N}$