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Differentiation of an Integral

  1. Oct 27, 2004 #1
    I have this equation


    [tex]
    \int_t^T n(s) (1- e^{-c (T-s)}) ds = c F(T)
    [/tex]


    and I need to differentiate both sides with respect to T

    [tex]
    \frac{\partial }{\partial T}
    [/tex]

    to get the following result

    [tex]
    \int_t^T n(s) ( e^{-c (T-s)}) ds = \frac{\partial F(T)}{\partial T}
    [/tex]

    How was it done ? What integration and differentiation rule was used ? If you could show it step by step I would appreciate.
     
  2. jcsd
  3. Oct 27, 2004 #2

    shmoe

    User Avatar
    Science Advisor
    Homework Helper

    I would rewrite the integral:

    [tex]
    \int_t^T n(s) (1- e^{-c (T-s)}) ds = \int_t^T n(s)ds-e^{-cT}\int_t^T n(s) e^{cs}} ds
    [/tex]

    Then use the tried and true fundamental theorem of calculus (assuming g is continuous):

    [tex]\frac{d}{dT}\int_{a}^{T}g(s)ds=g(T)[/tex]

    The purpose of rewriting was to remove any potentially confusing dependance of T from the integrands.
     
  4. Oct 27, 2004 #3
    Yeah, sure. Now I see it.

    Thanks very much for the prompt reply
     
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