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Math Help - Most general antiderivative of a function

  1. #1
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    Most general antiderivative of a function

    Is my answer correct?

    g(x) = (5-4x^3 + 2x^6) / x^6

    g'(x) = (5 - 4x^3 + 2x^6) (x^-6) + C

    g'(x) = (5-(4x^4)/4 + (2x^7)/7) (x^-5)/-5 + C

    g'(x) = (5 - x^4 + 2/7x^7) / (5x^5) + C

    g'(x) = (-x^4 + 2/7x^7) / (x^5) + C
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  2. #2
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    There is at least one mistake per line, both algebra and calculus errors. Try this:

    g(x)=(5-4x^{3} + 2x^{6}) / x^{6}=5x^{-6}-4x^{-3}+2.

    So far, I haven't done any calculus whatsoever, and thus there should be no constant of integration yet. In addition, the title of your post indicated that you are taking an antiderivative. It puzzled me, then, when I saw the notation g'(x), which is the notation for a derivative.

    In any case, you can now integrate term-by-term. What do you get?
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  3. #3
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    Thanks for your reply Ackbeet. You are right, I am trying to integrate g(x) = (5-4x^3 + 2x^6) / x^6 and should not have used g prime.

    Your hint was very helpful. I reworked the sum:

    g(x) = (5-4x^3 + 2x^6) / x^6

    = (5/x^6) - [(4x^3)/x^6] + [(2x^6)/x^6] + C

    = (5x^-6) - (4x^3)(x^-6) + (2x^6)(x^-6) + C

    = (5x^-6) - (4x^-3) + 2 + C

    = [(5x^-5)/-5] - [(4x^-2)/-2] + 2x + C

    = (-x^-5) + (2x^-2) + 2x + C

    = (-1/x^5) + (2/x^2) + 2x + C

    Is this correct?
    Last edited by sparky; January 26th 2011 at 04:32 PM. Reason: I left out the constant C
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  4. #4
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    Quote Originally Posted by sparky View Post
    Thanks for your reply Ackbeet. You are right, I am trying to integrate g(x) = (5-4x^3 + 2x^6) / x^6 and should not have used g prime.

    Your hint was very helpful. I reworked the sum:

    g(x) = (5-4x^3 + 2x^6) / x^6

    = (5/x^6) - [(4x^3)/x^6] + [(2x^6)/x^6] + C
    NO C! you don't have an added constant of integration until after you integrate! Ackbeet told you that.

    = (5x^-6) - (4x^3)(x^-6) + (2x^6)(x^-6) + C

    = (5x^-6) - (4x^-3) + 2 + C

    = [(5x^-5)/-5] - [(4x^-2)/-2] + 2x + C
    This is the first line on which "+ C" should appear. But you shouldn't have "= " in front of it. That means it is equal to the same as all the previous lines, g(x), and it isn't.
    You should have
    \int g(x) dx= -x^{-5}-+ 2x^{-2}+ 2x+ C


    = (-x^-5) + (2x^-2) + 2x + C

    = (-1/x^5) + (2/x^2) + 2x + C

    Is this correct?
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  5. #5
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    Thanks alot HallsofIvy!
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