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Integration by parts involving an unknown function

  1. Apr 8, 2012 #1
    1. The problem statement, all variables and given/known data

    I have attached a picture including 2 equations: (2.13) and (2.14)
    I don't understand how they got from (2.13) to (2.14) using integration by parts

    2. Relevant equations



    3. The attempt at a solution
    For the integral:
    [itex]\int_{\tau_0}^t\sigma(\tau)d\tau= \left[\sigma(\tau)\tau\right]_{\tau_0}^{t}-\int_{\tau_0}^t\dot{\sigma}(\tau)\tau d\tau[/itex]
    by assuming [itex]u=\sigma(\tau)[/itex] and [itex]v'=1[/itex] for integration by parts.
    This gives:
    [itex]\sigma(t)t-\sigma(\tau_0)\tau_0-\int_{\tau_0}^{t}\tau\dot{\sigma}(\tau)d\tau[/itex]

    So, even when we do combine this result with the other terms in equation (2.13), I don't understand how (2.14) does not depend on [itex]\sigma(t)[/itex].

    Thank you in advance for any ideas.
     

    Attached Files:

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  3. Apr 8, 2012 #2

    I like Serena

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    Homework Helper

    Hi sara_87! :)

    One of your terms is ##\sigma(t)t##.
    You can rewrite that as an integral:

    $$t \sigma(t)+C=t \int_{\tau_0}^t \dot\sigma(\tau)d\tau=\int_{\tau_0}^t t \dot\sigma(\tau)d\tau$$

    The variable t can be moved into the integral, since you do not integrate over t, so t behaves like a constant with respect to integration.
     
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