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A Integration by parts of a differential

  1. Jul 28, 2017 #1
    I'll cut the long story short. What on earth happened here:

    integration.png

    I seem to be unable to do the integration by parts of the first term. I end up with a lot of dx's.
     
  2. jcsd
  3. Jul 28, 2017 #2

    Mark44

    Staff: Mentor

    Try splitting the integral into two parts, with this as the first one:
    ##\int_{x_i}^{x_j} \frac d{dx}\left( kA\frac{dT}{dx}\right)N_i(x)dx##
     
  4. Jul 28, 2017 #3
    I do know how to split the integral. I just wonder what happened to the identity. Judging from the result, as in the format int(u dv) = uv - int(v du);
    u = d/dx (kA dT/dx)
    dv = Ni(x)dx

    The problem maybe is that I don't know how to calculate du. What would be the result?
     
  5. Jul 28, 2017 #4
    Because we're working with nice single-variable functions (presumably), [tex]dq=\frac{dq}{dx} \, dx=\frac{d}{dx}(q) \, dx[/tex] To me, it looks like they let [tex]dv=\frac{d}{dx}\left(kA \frac{dT}{dx}\right) \, dx[/tex] and [tex]u=N_i(x)[/tex] This would imply [tex]v=kA \frac{dT}{dx}[/tex] and [tex]du=dN_i=\frac{dN_i}{dx} \, dx[/tex] which appears to be consistent with the result they got.
     
  6. Jul 28, 2017 #5

    Mark44

    Staff: Mentor

    What happens if you instead let ##u = N_i(x)## and ##dv = \frac d {dx} \left(kA \frac{dT}{dx}\right) dx##? Finding v shouldn't be that difficult.
     
  7. Jul 28, 2017 #6
    Thanks!
     
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