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Math Help - differentiating under the integral sign

  1. #1
    Super Member Random Variable's Avatar
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    differentiating under the integral sign

    To prove that  \frac {d}{dt} \int^{b}_{a} f(x,t) \ dx = \int^{b}_{a} \frac{d}{dt} f(x,t) \ dx at  t=t_{0}

    is it sufficient to show that f(x,t) and \frac{d}{dt}f(x,t) are continous for x in the range of integration and for t in a interval about t_{0} ? And if you want to differentiate under the integral sign multiple times, is it sufficient to show that each derivative of f(x,y) is continous for  x in the range of intergration and for  t in a interval about t_{0}?


    And what about improper integral  \frac {d}{dt} \lim_{b \to \infty} \int^{b}_{a} f(x,t) \ dx ?

    Is it sufficient to show in addition that  |f(x,t)| and \Big|\frac{d}{dt}f(x,t)\Big| are bounded above by functions independent of t and the integrals of both of those functions converge over the interval [a, \infty) ?
    Last edited by Random Variable; July 30th 2009 at 07:15 PM.
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    Super Member Random Variable's Avatar
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    I must ask the worst questions.
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  3. #3
    Moo
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    Hello,

    Leibniz integral rule - Wikipedia, the free encyclopedia

    Is it what you're looking for ? (link to the proof in wikipedia)
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  4. #4
    Super Member Random Variable's Avatar
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    Quote Originally Posted by Moo View Post
    Hello,

    Leibniz integral rule - Wikipedia, the free encyclopedia

    Is it what you're looking for ? (link to the proof in wikipedia)
    Yes. Thanks.

    But I'm still uncertain about improper integrals.
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