: Integral of (n+1)th derivative

[kfdleb]

URGENT: Integral of (n+1)th derivative

Homework Statement

let f⁽ⁿ⁺¹⁾ be integrable on [a;b]; show that

f(b)=$$\sum$$ $$\frac{f^(r)(a)}{r!}$$(b-a)^r +$$\frac{1}{n!}$$ $$\int^{a}_{b}$$f⁽ⁿ⁺¹⁾(t)(b-t)ⁿdthint:integrate by parts and use inductionPLEASE any idea about how to solve it would be really appreciated... I've been trying for more than an hour but no idea

 

[snipez90]

Well use the hint. Induct on n. For n = 1, show the equation holds by computing the integral that you get on the RHS after setting n = 1. This is fairly straightforward.

 

[Dick]

Start by actually doing the integration by parts. Treat the integral as u*dv where u=f^(n+1)(t) and dv=(b-t)^n*dt. Once you've got that straight then start worrying about the induction.

 

[kfdleb]

10x a lot