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Fundamental Theorem of Calculus

  1. Nov 13, 2008 #1
    1. The problem statement, all variables and given/known data
    The FTC states that
    [tex]{d\over{dx}}\int_a^x f(t)dt = f(x)[/tex]
    Now, how do I do something like
    [tex]{d\over{dx}}\int_a^{g(x)} f(t)dt = ?[/tex]

    2. Relevant equations

    3. The attempt at a solution
    I know that it has to do with the chain rule, but I forgot my textbook at school and I can't seem to find it online (e.g. Wikipedia).
  2. jcsd
  3. Nov 13, 2008 #2


    Staff: Mentor

    Changing the variable to u, where u = g(x), the integral looks like this:
    [tex]{d\over{du}}\int_a^u f(t)dt = f(u)[/tex]

    The trouble is, you want [tex]{d\over{dx}}\int_a^u f(t)dt = f(u)[/tex]

    So if you want d/dx(H(u)), that's the same as d/du(H(u))*du/dx, isn't it? (Here, H(u) represents the value of the definite integral."
  4. Nov 14, 2008 #3
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