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Math Primitives

  1. Mar 16, 2014 #1
    When we took the integral of a function f, the result is: [tex]\int f(x)dx = F(x) + C[/tex] But, F(x) + C can be rewritten like: [tex]F(x) + C = \mathcal{F}(x)[/tex] So, my first ask is: which is the name given for ##F(x)## and for ##\mathcal{F}(x)## ? I can't call both of primitive of f, because it's confuse. I already heard the term "family of antiderivative", I think that this term is the name of ##\mathcal{F}(x)##, thus, maybe, primitive is better for ##F(x)##. What do you think?

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    Second question: by FTC: [tex]\int_{x_0}^{x}f(x)dx = F(x) - F(x_0)[/tex] implies that: [tex]F(x) = \int_{x_0}^{x}f(x)dx + F(x_0)[/tex] This result, F(x), represents the ##F(x)## or the ##\mathcal{F}(x)## of the my 1st question?

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    OBS: if you affirm that F(x) of the 2nd question is equal to ##\mathcal{F}(x)## this implies that ##\int_{x_0}^{x}f(x)dx = F(x)## and ##F(x_0)=C## .
     
    Last edited: Mar 16, 2014
  2. jcsd
  3. Mar 16, 2014 #2

    Mark44

    Staff: Mentor

    What's the point? Is there some purpose in writing F(x) + C as ##\mathcal{F}(x)##?
    F is an antiderivative of f. I've also seen it called a primitive.
    F is any antiderivative of f (or any primitive of f).
     
  4. Mar 17, 2014 #3
    There is a importnat difference between ##F(x)## and ##\mathcal{F}(x)##, the 1st is a particular solution for ##\int f(x) dx## and the 2nd is general solution. By be different solutions, maybe they have different names. How I already heard a lot times the term "antiderivative" and "family of antiderivative", I asked if those terms are the names for ##F(x)## and for ##\mathcal{F}(x)## ...

    EDIT: YEAH! An teacher confirmed my hypothesis above.
     
    Last edited: Mar 17, 2014
  5. Mar 17, 2014 #4

    Mark44

    Staff: Mentor

    I disagree. The expression F(x) + C doesn't represent a particular antiderivative unless C is somehow specified to be a particular value. In that sense (i.e., C being an unspecified arbitrary value) F(x) + C represents the entire family of antiderivatives. Again, I still don't see the point of writing both F(x) + C and ##\mathcal{F}(x)##. IMO it's much ado about a minor point.
    I still disagree.
    For example, consider ##\int x^2 dx##.
    One antiderivative is (1/3)x3 + 7.
    All antiderivatives have the form (1/3)x3 + C.
     
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