How to prove that the

Question

{The integral with limit a to b *(the integral with limit x to b f(t)dt}dx=integral with limit a to b (x-a)f(x)dx

Help..

Answer 1

I'll get you started. Suppose
F(x) = ∫f(x) dx

∫(x-a)f(x) dx = ∫xf(x) dx - ∫af(x) dx = ∫xf(x) dx - aF(x)
Now, using integration by parts, let
u = x, du = dx
dv = f(x) dx, v = F(x)
∫xf(x) dx = xF(x) - ∫F(x) dx

∫[x,b] f(t) dt = F(b)-F(x)

see what you can do from here.

Answer 2

What is the function of f(b)?

Answer 3

doesn't matter. Just expand the expressions, and there will be lots of F(x). F(b), F(a)'s floating around. It will be the same on both sides of the equation