Derivation Of Multiple Integrals Depending On A Parameter

lion1977 · June 4th 2006, 06:51 AM

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

lion1977

**Rebesques** · August 11th 2006, 11:22 AM

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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