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Proving Integral of a Product

  1. Nov 30, 2006 #1
    1. The problem statement, all variables and given/known data

    Prove that
    2. Relevant equations
    Basically, the question also tells you that you should somehow use the fundamental theorem of calculus.

    3. The attempt at a solution
    This is pretty much asking me to prove that the derivative of f(x)g(x) is just f'(x)g(x) + f(x)g'(x)...and then I could use the fundamental theorem of calculus to say that the integral of the derivative is just the function itself.

    It seems pretty self-explanatory to me, so I havn't got a clue on where to start on this proof, I would appreciate a tip or two to help me get started:rolleyes:

    here's what I got so far. I don't know if this is right or wrong, so I would appreciate some comment on it

    [tex]let G(x)=f(x)g(x)[/tex]
    [tex]then G'(x)=f'(x)g(x)+f(x)g'(x)[/tex]

    [tex]\therefore f(x)g(x)-f(0)g(0) = G(x) - G(0)[/tex]
    [tex]\int^x_0 f'(t)g(t)dt+f(t)g'(t)dt[/tex]
    [tex]=\int^x_0 G'(t)dt[/tex]
    fundamental theorem of calculus states that:
    [tex]\int^x_a G'(t)dt = G(x)|^x_a[/tex]
    [tex]\therefore \int^x_0 G'(t)dt = G(x)|^x_0=G(x)-G(0) = f(x)g(x)-f(0)g(0)[/tex]

    sorry if the equations are not aligned properly, I'm a newbie at LaTex, thx all.
    Last edited: Nov 30, 2006
  2. jcsd
  3. Nov 30, 2006 #2


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    Homework Helper

    Well, I think that's pretty much all you need to do.

    Just as a detail on how to write it out, I would leave out the 3rd line "therefore f(x)g(x) ....." because you the same thing again in the last line, and it comes straight from your definition of the function G so it's pretty obvious.

    When you say "therefore" on line 3 it might make a reader think that line 3 follows from line 2, which is misleading.

    The key point (as you said) is to notice that f(x)g'(x) + f'(x)g(x) is the derivative of f(x)g(x). The rest is just notation, really.
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