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Math Help - Adding and subtracting integrals

  1. #1
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    Adding and subtracting integrals

    So, I'm really rusty with math and I have a question that is stumping me. It's making me really mad because I'm sure somewhere in the back of my mind I know how to do it, but I can't figure it out. Here is the question:

    I will denote: squared as (f)^2 and integrals as int[function]. all integrals go from a to b and f''(x) = second derivative of f

    Verify that for f,g in R:

    int[f''(x)^2] dx - int[g''(x)^2] dx = int[(f''(x)-g''(x))^2] dx + 2 int[g''(x)(f''(x)-g''(x))]dx



    I'd really appreciate some help or ideas, I'm pretty sure there is a way to do it without trying to solve the integral, and all i have to do is verify the result. thanks!
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  2. #2
    A Plied Mathematician
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    Re: Adding and subtracting integrals

    So you're trying to verify that

    \int_{a}^{b}[f''(x)]^{2}\,dx-\int_{a}^{b}[g''(x)]^{2}\,dx
    =\int_{a}^{b}[f''(x)-g''(x)]^{2}\,dx+2\int_{a}^{b}g''(x)[f''(x)-g''(x)]\,dx.

    Is that correct? If so, I'd say your starting-point would be to note that

    \int_{a}^{b}f(x)\,dx+\int_{a}^{b}g(x)\,dx=\int_{a}  ^{b}(f(x)+g(x))\,dx, assuming all three integrals exist. What does this enable you to do?
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  3. #3
    MHF Contributor Siron's Avatar
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    Re: Adding and subtracting integrals

    In my opinion it's just a fact of using some algebraic rules.
    If you want to prove it: RHS -> LHS:
    \int_{a}^{b} [f''(x)-g''(x)]^2dx +2\int_{a}^{b}g''(x)[f''(x)-g"(x)]dx
    =\int_{a}^{b} ([f''(x)]^2-2[f''(x)g''(x)]+[g''(x)]^2)dx +2\int_{a}^{b} (g''(x)f''(x)-[g''(x)]^2)dx
    =\int_{a}^{b} ([f''(x)]^2-2[f''(x)g''(x)]+[g''(x)]^2+2g''(x)f''(x)-2[g''(x)]^2)dx

    Simplify ...
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  4. #4
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    Re: Adding and subtracting integrals

    Quote Originally Posted by seamstress View Post
    S
    Verify that for f,g in R:
    int[f''(x)^2] dx - int[g''(x)^2] dx = int[(f''(x)-g''(x))^2] dx + 2 int[g''(x)(f''(x)-g''(x))]dx
    I think that you have entered some terms incorrectly.
    Should it be
    \int {\left( {f''} \right)^2 }  - \int {\left( {g''} \right)^2 }  = \int {\left( {f'' - g''} \right)^2 }  - 2\int {f''g''} ~?
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  5. #5
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    Re: Adding and subtracting integrals

    no, the two up there had it. thanks though!
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  6. #6
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    Re: Adding and subtracting integrals

    yeah, now i really feel like an idiot...that was super easy. guess i really am rusty. thank you
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  7. #7
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    Re: Adding and subtracting integrals

    however, i still can't figure out how to integrate the equation by parts. any pointers? thanks
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  8. #8
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    Re: Adding and subtracting integrals

    NEVERMIND i got it. just please don't answer any more of my questions today- it's been an off one.
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