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Differentiating integrals

  1. Jan 30, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex]d/dx[/tex] ([tex]\int[/tex][tex]^{x}_{0}[/tex] x2t2dt)
    So the problem is to solve the derivative of the integral [tex]\int[/tex] x2t2dt from 0 to x.

    2. Relevant equations

    [tex]d/dx[/tex] ([tex]\int[/tex][tex]^{x}_{a}[/tex] f(t)dt) = f(x)

    3. The attempt at a solution

    I'm really unsure of how this should be computed but this was my guess:

    [tex]d/dx[/tex] ([tex]\int[/tex][tex]^{x}_{0}[/tex] x2t2dt) = [tex]d/dx[/tex] ([tex]1/3[/tex]x2(x)3 -([tex]1/3[/tex]x2(0)3)) = [tex]d/dx[/tex] ([tex]1/3[/tex]x5) = [tex]5/3[/tex]x4

    So, first I calculated the integral with respect to t and then derivated it with respect to x. But it feels wrong. I don't know how to treat the function x2t2 because the variable x is both part of the function and an endpoint of the interval for integration.
  2. jcsd
  3. Jan 30, 2009 #2


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    Homework Helper

    You don't have to treat x^2 as 'part of the function'. It's just a constant multiplying the function t^2. You can take it out of the integral. If you have to do more complicated problems where the x and t are mixed up so you can't do that, check out the Leibniz Integral Rule. So your answer is correct.
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