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Is this notation correct?

  1. Feb 25, 2012 #1
    1. The problem statement, all variables and given/known data
    I need to plug 0 into an anti-derivative of f, and I'm wondering whether it can be written as:

    [itex]y = \int f(0)dx[/itex]

    or do I HAVE to write it as:
    [itex]F(x) =\int f(x)dx[/itex]
    [itex]y = F(0)[/itex]

    2. Relevant equations
    See 1.


    3. The attempt at a solution
    See 1.
     
  2. jcsd
  3. Feb 25, 2012 #2

    lanedance

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    neither really, as the variable you integrate over is treated as a dummy variable, so really something like below would be better
    [tex]
    F(x)=\int_{x_0}^{x}f(s)ds
    [/tex]

    the x0 point determines your constant of integration
     
    Last edited by a moderator: Feb 25, 2012
  4. Feb 25, 2012 #3
    Why can't it just be:
    [itex]F(x) + C =\int f(x)dx[/itex]

    [itex]y = F(0) + C[/itex]

    With "C" being the constant of integration.
     
  5. Feb 25, 2012 #4

    Fredrik

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    When I see the expression ##\int f(0) dx##, I'm thinking that f(0) is a constant that can be taken outside of the integral. The notation ##F(x)=\int f(x)dx## is no better, because the left-hand side depends on x (unless F is a constant function), while the right-hand side doesn't. So the equality doesn't define a function, even if you specify that it holds for all x.

    Why not just say something like
    Let F be any function such that F'=f.​
    or
    Let F be the unique function such that F'=f and F(0)=C.​
    ?
    Then you can denote the value of F at 0 by F(0).

    Lanedance's suggestion is also good.
     
  6. Feb 26, 2012 #5

    lanedance

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    yeah just to add the integral depends on only on the endpoints of the integral, and the form of the function.

    when you have f(x) and sat the antiderivative is F(x) + c, the easiest way to think of it is the constant is related to where the integral started (x0) whilst x represents the end of the integration interval. In performing the integration, f takes all possible values on the interval (x0,x)
     
  7. Feb 26, 2012 #6
    Does one have to add the constant of integration (C) when using the 'from 0 to x' definite integral method to express indefinite integrals at a given x?

    Ex:
    [tex]F(x)=\int_{0}^{x} f(t)dt[/tex]

    I still have to add "C" to F(x), am I right?
     
  8. Feb 26, 2012 #7

    Fredrik

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    If the problem you're trying to solve asked you to find all antiderivatives of f, then yes. The F defined by the formula above satisfies F'=f and F(0)=0, so it's just one of the infinitely many antiderivatives of f. However, note that ##\int_a^x f(t)dt## is equal to ##\int_0^x f(t)dt## plus a constant that depends on a. So each choice of a gives you a different antiderivative of f.
     
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