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Math Help - integration by parts problem

  1. #1
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    integration by parts problem

    Suppose that f(x) and g(x) are functions with f(0)=g(0)=0, f(1)=g(1)=0, and with continuous second derivatives.

    Use integration by parts to show that

    <br /> <br />
\int f''(x)g(x)\,dx = \int f(x)g''(x)\,dx<br />

    Then come up with specific examples for f(x) and g(x) that satisfy the above.

    I tried using
    u=g(x) v=f(x)
    du=g'(x)dx dv=f''(x)dx

    then

    u=g'(x) v=f(x)
    du=g''(x)dx dv=f'(x)dx

    but I ended up with

    <br /> <br />
\int f''(x)g(x)\,dx = g(x)f'(x)-g'(x)f(x)- \int f(x)g''(x)\,dx<br /> <br />

    and I don't know how to get rid of the two middle functions...

    if I switched what U was for the second time I applied it, I get back to where I began and If I apply IBP again, i get into 3rd derivatives...

    I'm really confused. Can someone give me some advice?
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  2. #2
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    Quote Originally Posted by valkyrie View Post
    Suppose that f(x) and g(x) are functions with f(0)=g(0)=0, f(1)=g(1)=0, and with continuous second derivatives.

    Use integration by parts to show that

    <br /> <br />
\int f''(x)g(x)\,dx = \int f(x)g''(x)\,dx<br />

    Then come up with specific examples for f(x) and g(x) that satisfy the above.

    I tried using
    u=g(x) v=f(x)
    du=g'(x)dx dv=f''(x)dx

    then

    u=g'(x) v=f(x)
    du=g''(x)dx dv=f'(x)dx

    but I ended up with

    <br /> <br />
\int f''(x)g(x)\,dx = g(x)f'(x)-g'(x)f(x)- \int f(x)g''(x)\,dx<br /> <br />

    and I don't know how to get rid of the two middle functions...

    if I switched what U was for the second time I applied it, I get back to where I began and If I apply IBP again, i get into 3rd derivatives...

    I'm really confused. Can someone give me some advice?
    I am guessing that the integrals have limits of integration, and the they are from 0 to 1 or the statement is not true.

    dont forget to evaluate the middle parts and the conditions that f and g are = 0 at 1 and 0 will make them go away.
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  3. #3
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    Hmmm, thanks. they are from 0 to 1 but I couldnt get the BB code to work properly!

    I don't really remember in such an abstract situation... how do I evaluate the middle parts? do I just set them from 0 to 1?
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  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by valkyrie View Post
    Hmmm, thanks. they are from 0 to 1 but I couldnt get the BB code to work properly!

    I don't really remember in such an abstract situation... how do I evaluate the middle parts? do I just set them from 0 to 1?
    g(x)f'(x)\bigg|_{0}^{1}-g'(x)f(x)\bigg|_{0}^{1}=g(1)f'(1)-g(0)f'(0)-(g'(1)f(1)-g'(0)f(0))

    but since f(0)=f(1)=g(0)=g(1)=0

    g(1)f'(1)-g(0)f'(0)-(g'(1)f(1)-g'(0)f(0))=0f'(1)-0f'(0)-g'(1)0+g'(0)0=0

    This is what we wanted yay
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  5. #5
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    Quote Originally Posted by TheEmptySet View Post
    g(x)f'(x)\bigg|_{0}^{1}-g'(x)f(x)\bigg|_{0}^{1}=g(1)f'(1)-g(0)f'(0)-(g'(1)f(1)-g'(0)f(0))

    but since f(0)=f(1)=g(0)=g(1)=0

    g(1)f'(1)-g(0)f'(0)-(g'(1)f(1)-g'(0)f(0))=0f'(1)-0f'(0)-g'(1)0+g'(0)0=0

    This is what we wanted yay
    OH. wow. okay thank you that makes A LOT of sense. I was having an idiot moment.
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