1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Real Analysis: Integration

  1. Oct 29, 2008 #1
    1. The problem statement, all variables and given/known data
    Let f be of class C1 on [a, b], with f(a) = f(b) = 0. Show that [tex]\int_a^b xf(x)f'(x)dx[/tex] = [tex]-1/2 \int_a^b [f(x)]^2 dx[/tex].

    2. Relevant equations
    If F is an antiderivative of f, then [tex]\int_a^b f(t)dt = F(b) - F(a)[/tex]

    3. The attempt at a solution
    I'm just really not sure how to begin this one. I know that because f is of class C1 that f' is continuous. Maybe change of variables?
  2. jcsd
  3. Oct 29, 2008 #2
    Use integration by parts!
  4. Oct 29, 2008 #3


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Less dogmatically... one of the major differences between the two things you're trying to show equal is that one has a derivative in it, and the other doesn't. And what methods do you know that can increase/decrease how derivated a part of your integrand is?
  5. Oct 29, 2008 #4
    [tex]\int_a^b xf(x)f'(x)dx[/tex]

    [tex] u=x, du=dx, v=\int f(x)f'(x)dx[/tex]

    [tex] v=\int f(x)f'(x)dx, t=f(x), dt=f'(x)dx=>\int tdt=\frac{1}{2}t^2=\frac{1}{2}[f(x)]^2[/tex]

    [tex]\frac{1}{2}x[f(x)]|_a^b-\frac{1}{2}\int_a^b [f(x)]^2dx=-\frac{1}{2}\int_a^b [f(x)]^2dx[/tex]

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook