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Math Help - Differentiation under the integral sign

  1. #1
    MHF Contributor Mathstud28's Avatar
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    Differentiation under the integral sign

    Ok, could someone please explain this to me, for I have tried to find it on the internet but everywhere (especially the wikipedia) article is written from an analysis point of view. Functionally I would like to know how it works.

    I know that

    If I(\theta)=\int_a^{b}f(x,\theta)d\theta

    That by Lebniz

    \frac{dI(\theta)}{d\theta}=\int_a^{b}f(x,\theta)dx

    So now here is where I am in the dark, does this imply that lets say we integrate the right hand side and get

    F(\theta) (obviously there will be no more x's)[/tex]

    Then we have that

    I(\theta)=\int{F(\theta)}d\theta

    So

    Like

    \int_a^{b}f(x,5)dx=\int{F(\theta)}d\theta\bigg|_{\  theta=5}

    Please if anyone could enlighten me, I have been curious about this method for a while and cannot seem to find a comprehensible source. So if somone could explain it that would be great.


    Mathstud
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  2. #2
    Super Member wingless's Avatar
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    \int_a^b f(x,\theta)~d\theta is a function of x, not \theta.
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by wingless View Post
    \int_a^b f(x,\theta)~d\theta is a function of x, not \theta.
    If you are talking about my first integral that was a typo it should have been dx.
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  4. #4
    Super Member wingless's Avatar
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    I don't think I understand what you are asking.

    If I(\theta)=\int_a^{b}f(x,\theta)~d\theta,

    then \frac{dI(\theta)}{d\theta}=\int_a^{b}\frac{\partia  l}{\partial \theta}f(x,\theta)~dx

    Can you fix the typos and ask it more clearly?
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by wingless View Post
    I don't think I understand what you are asking.

    If I(\theta)=\int_a^{b}f(x,\theta)~d\theta,

    then \frac{dI(\theta)}{d\theta}=\int_a^{b}\frac{\partia  l}{\partial \theta}f(x,\theta)~dx

    Can you fix the typos and ask it more clearly?
    Of course...I am asking about the method of integrating definite integrals.

    So say you have an an integral and you set it up so it looks like this

    I(\theta)=\int_a^{b}f(x,\theta)dx

    where theta is a fixed constant

    I believe that

    \frac{dI(\theta)}{d\theta}=\int_a^b\frac{\partial{  f(x,theta)}}{\partial{\theta}}dx

    So now if we inegrated the right side we would get a function of theta

    So lets call that function F(\theta)

    So now we have that

    \frac{dI(\theta)}{d\theta}=F(\theta)

    But since I(\theta) was our original quantity we wanted to find

    I suppose (and this is where I am unsure)

    That you then compute \int{F(\theta)}d\theta

    Because

    I(\theta)=\int{F(\theta)}d\theta

    If so, where does the constant of integration go?
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