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Math Help - Integration with Respect to a Derivative

  1. #1
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    Integration with Respect to a Derivative

    In general, what is the integral \int_{0}^{\infty} f^n(z) dn where n is the nth derivative of f(z)?
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  2. #2
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    your answer is probably 0 because n is a constant. integration with respect to a constant(ie n) is 0,isn't it???
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by PaulDirac2 View Post
    In general, what is the integral \int_{0}^{\infty} f^n(z) dn where n is the nth derivative of f(z)?
    I mean? What do you expect? If this function is integrable, which it has to be, I would say that it equals \lim_{z\to\infty}\left[f^{n-1}(z)-f^{n-1}(0)\right]

    Quote Originally Posted by Pulock2009 View Post
    your answer is probably 0 because n is a constant. integration with respect to a constant(ie n) is 0,isn't it???
    No. Read more carefully, it is not a constant function.
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    Interation with Respect to a Derivative

    In the integral I posted z is not the variable that I wanted to integrate with respect to.
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    Quote Originally Posted by PaulDirac2 View Post
    In general, what is the integral \int_{0}^{\infty} f^n(z) dn where n is the nth derivative of f(z)?
    What, exactly, does f^n(z) mean? Is it the nth power of f or the nth derivative (more commonly written f^{(n)}(z))? If it means the nth derivative, and you are integrating with respect to its nth derivative that is exactly the same as \int x dx= (1/2)x^2 evaluated at f^{(n)}(z), \frac{1}{2}\left[\lim_{x\to\infty} (f^{(n)}(x))^2- f^{(n)}(0)\right].

    If you mean f to the power n, then it will depend more strongly on exactly what f is.
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  6. #6
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    Thumbs down Z is a Constant

    In the integral, the only variable is n, the nth derivative. z is a constant.
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    Integration HELP

    I am really having trouble with the concept of this problem. If you integrated with respect to the variable n by taking the antiderivative of n, and the limits of the integral are set to 1 and 0, you come up with a 1/2 derivative which makes no sense at all. I cannot take the antiderivative of the function as a whole because I am not integrating with respect to z, since z is a constant. If anyone could point me in the right direction I would appreciate it very much.
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