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Prove the following integral

  1. Nov 17, 2014 #1
    1. The problem statement, all variables and given/known data
    prove that the ∫01 f(x) = ∫01f(1-x)

    2. Relevant equations


    3. The attempt at a solution
    I got all the way to ∫01-f(u) du where u = 1-x but I don;t know how to prove it.
     
  2. jcsd
  3. Nov 17, 2014 #2

    ehild

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    When you substitute the integration variable the limits also change.
     
  4. Nov 17, 2014 #3
    Okay, so -∫1 f(u) du = ∫01f(u) du. but how do i relate this to ∫f(x)?
     
  5. Nov 17, 2014 #4

    ehild

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    It does not matter how do you name the integration variable. It can be x instead of u. The function is the same, and so are the limits.
     
  6. Nov 17, 2014 #5

    Ray Vickson

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    Don't you see that ##\int_0^1 f(x) \, dx = \int_0^1 f(u) \, du = \int_0^1 f(\text{anything}) \, d\text{anything}##?
     
  7. Nov 17, 2014 #6

    Mark44

    Staff: Mentor

    To elaborate on what Ray said, I have added variables in the limits of integration to emphasize that in each integral we have a different dummy variable.
    $$ \int_{x = 0}^1 f(x) \, dx = \int_{u = 0}^1 f(u) \, du = \int_{\text{anything} = 0}^1 f(\text{anything}) \, d\text{anything}$$

    When you use substution to change the variable of integration, you need to either change the limits of integration (which was not done in the above) or undo the substitution.
     
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