Asked by Buh
How to prove that the
{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..
{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..
Answers
Answered by
Steve
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.
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.
Answered by
Buh
What is the function of f(b)?
Answered by
Steve
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
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