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The fundamental theorem of calculus

  1. Sep 10, 2010 #1
    Can we apply the fundamental theorem of calculus to an integrand that's a function of the differentiating variable?

    For example, can one still use the fundamental theorem of calculus when trying to find the derivative of this function:

    [tex]f(x) = \int_{0}^{x} \frac{cos(xt)}{t}dt[/tex]
     
  2. jcsd
  3. Sep 10, 2010 #2

    Hurkyl

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    Nope. But the idea of the proof can still be used -- you can derive a general formula for
    [tex]\int_{f(x)}^{g(x)} h(x, t) \, dt[/tex]
    with suitable conditions on the three functions.
     
  4. Sep 10, 2010 #3
    Well, here's my guest for the formula:

    If

    [tex]H(x) = \int_{f(x)}^{g(x)} h(x, t) \, dt[/tex],

    then

    [tex]H'(x) = h(x,g(x)) - h(x,f(x))[/tex]
     
  5. Sep 10, 2010 #4

    Hurkyl

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    Have you tried not guessing, and actually trying to work it out, starting with the very definition of derivative?
     
  6. Sep 10, 2010 #5
    Let

    [tex]F(x) = \int_{a}^{g(x)} f(x,t) dt[/tex]


    Now define

    [tex]I(x,y) = \int f(x,t) dt [/tex]

    with t = y after indefinite integration over t.

    Then

    [tex] F'(x) = I_x(x,g(x)) - I_x(x,a) + I_x(x,g(x))g'(x)[/tex]
     
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