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Differentiation of an Integral

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

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

    and I need to differentiate both sides with respect to T

    \frac{\partial }{\partial T}

    to get the following result

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

    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


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    I would rewrite the integral:

    \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

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


    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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