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Math Help - Find F'(x) and F''(x)

  1. #1
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    Find F'(x) and F''(x)

    Let f be everywhere continuous and set
    F(x) = \int _0\,^x\![t \int _1\,^t\!f(u)du] dt
    Find F'(x) and F''(x)

    How do I deal with the double interal? I have \int _0\,^x\![t f(t)] dt, is it the right first step?
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  2. #2
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    Yes, that's the first step.

    Now is just a matter of simple application of the FTC.
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  3. #3
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    Quote Originally Posted by Krizalid View Post
    Yes, that's the first step.

    Now is just a matter of simple application of the FTC.

    So, is the answer F'(x) = xf(x), then what is F''(x)?
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  4. #4
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    Quote Originally Posted by 450081592 View Post
    Let f be everywhere continuous and set
    F(x) = \int _0\,^x\![t \int _1\,^t\!f(u)du] dt
    Find F'(x) and F''(x)

    How do I deal with the double interal? I have \int _0\,^x\![t f(t)] dt, is it the right first step?
     F(x) = \int _0^x\left[t \int _1^t f(u)~du \right] ~dt

    let G(t) = t \int _1^t f(u)~du, then we have

    F(x) = \int _0^x G(t) ~dt.

    By FTC,

    F'(x) = G(x)

    and F''(x) = \frac{d}{dx} \left(x \int _1^x f(u)~du \right)= \int _1^x f(u)~du + xf(x)
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