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Suppose in general a pair of functions

  1. Feb 24, 2005 #1
    Suppose in general that we have two functions

    [tex]

    F(x)= \int_{0}^{cos x}e^{xt^2} dt
    [/tex]
    [tex]
    G(x)= \int_{0}^{cos x}\(t^2e^{xt^2} dt
    [/tex]
    [tex]
    H(x) = G(x) - F'(x)
    [/tex]

    Where, I need to prove that
    [tex]
    H(\frac{\pi}{4}) = e^\frac{\pi}{8}/\sqrt{2}
    [/tex]

    Okay, so far I have computed the integrals of both of these functions, where I am confused is when computing [tex] F'(x) [/tex] do I differentiate the integrand with respect to x only, and then simply subtract the two functions. Sorry for the edit, I left off the [tex] dt [/tex] for both integrals. Any help would be appreciated!!
     
    Last edited: Feb 24, 2005
  2. jcsd
  3. Feb 24, 2005 #2

    NateTG

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    These integrals are a bit confusing. Are they supposed to be, for example:
    [tex]F(x)= \int_{0}^{cos x}e^{xt^2} dt[/tex]

    Or something different?
     
  4. Feb 24, 2005 #3
    Yes I corrected the original post, sorry I left off the [tex] dt [/tex] for both integrals.
     
  5. Feb 25, 2005 #4

    HallsofIvy

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    To find the derivative, with respect to x, of [itex]F(x)= \int_{0}^{cos x}e^{xt^2}dt [/itex], use "LaGrange's Formula" [tex]\frac{d\int_{a(x)}^{b(x)} f(x,t)dt}{dx}= \int_{a(x)}^{b(x)} \frac{\partial f(x,t)}{\partial x} dt+ F(b(x))\frac{db}{dx}- F(a(x)\frac{da}{dx}[/itex].
     
  6. Feb 25, 2005 #5

    arildno

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    You did mean Leibniz' formula, HallsofIvy?
     
  7. Feb 25, 2005 #6

    HallsofIvy

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    I am always making that mistake. Do you suppose I could convince them to swap names?
     
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