# Integration by parts involving an unknown function

1. Apr 8, 2012

### sara_87

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:
$\int_{\tau_0}^t\sigma(\tau)d\tau= \left[\sigma(\tau)\tau\right]_{\tau_0}^{t}-\int_{\tau_0}^t\dot{\sigma}(\tau)\tau d\tau$
by assuming $u=\sigma(\tau)$ and $v'=1$ for integration by parts.
This gives:
$\sigma(t)t-\sigma(\tau_0)\tau_0-\int_{\tau_0}^{t}\tau\dot{\sigma}(\tau)d\tau$

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 $\sigma(t)$.

Thank you in advance for any ideas.

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2. Apr 8, 2012

### I like Serena

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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