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Math Help - Derivation Of Multiple Integrals Depending On A Parameter

  1. #1
    lion1977
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    Derivation Of Multiple Integrals Depending On A Parameter

    Please, I need a help to derive the formula you can find in the attached file
    Many many thanks for the help

    lion1977
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  2. #2
    Super Member Rebesques's Avatar
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    A real mess!!!

    Let's see. Use the formula

    \frac{d}{dt}\left(\int_{\phi_1(t)}^{\phi_2(t)}F(t,  x)dx\right)
    =F(t,\phi_2(t))\frac{d\phi_2}{dt}(t)-F(t,\phi_1(t))\frac{d\phi_1}{dt}(t)+\int_{\phi_1(t  )}^{\phi_2(t)}\frac{\partial F}{\partial t}(t,x)dx

    to calculate the cases n=1,2,3 say. A pattern should emerge - and then do induction.
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