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Fundemental Theorem of Calculus

  1. Dec 11, 2009 #1
    1. The problem statement, all variables and given/known data
    Hi guys

    From the Fundemental Theorem of Calculus we know that if we have a continuous function f : [a,b] -> R and F is the function on (a,b) defined by
    [tex]
    F(x)=\int_a^xf(t)dt,
    [/tex]
    then F is differentiable on (a,b) with F'(x)=f(x) for all x in (a,b), i.e.
    [tex]
    \frac{d}{dx}\int_a^x f(t)dt=f(x).
    [/tex]


    Question: Is it correct also to write
    [tex]
    \frac{d}{dx}\int f(t)dt=f(x)?
    [/tex]

    If not, then is there a way of expressing [itex]\frac{d}{dx}\int_a^x f(t)dt=f(x)[/itex] without limits on the integral?
     
  2. jcsd
  3. Dec 11, 2009 #2
    When you write that, everyone knows what you mean -- but it's not correct. The reason is that
    [tex]\int f(t)\,dt[/tex]
    doesn't name a function of [tex]x[/tex], even though some dependence on [tex]x[/tex] is clearly intended. You could think of it as naming a one-parameter family of functions
    [tex]F_a(x) = \int_a^x f(t)\,dt[/tex]
    where the parameter [tex]a[/tex] is the lower limit of the integral. The parameter shows up in another guise as the customary "constant of integration" [tex]C[/tex], which is connected with the fact that [tex]F_b - F_a[/tex] is the constant function equal to
    [tex]\int_a^b f(t)\,dt[/tex].

    The functions [tex]F_a[/tex] all differ by a constant, so they have the same derivative, which is [tex]f[/tex] by the fundamental theorem. But the one-parameter family of functions [tex]\{F_a : a \in \mathbb{R}\}[/tex] isn't something you can use the same notation to differentiate.
     
  4. Dec 12, 2009 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Occaisionally, you will see the notation
    [tex]\int^x f(t)dt[/tex]
    (note the lack of a lower limit) to indicate the general anti-derivative.
     
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