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How to find the error in this particular integration process?

  1. Jan 20, 2013 #1
    1. The problem statement, all variables and given/known data
    The problem, more like a riddle, that our teacher gave it to us was to find the error in this particular equation:

    integral of 1/x dc = ln (absolute value of (x)) + C ; perfectly fine
    integral of dx/3x = 1/3 integral of dx/x ; also fine
    1/3 integral dx/x = 1/3 ln (absolute value of x) + C ; not sure from here
    integral of dx/3x ≠ 1/3 ln (absolute value of 3x) + C


    3. The attempt at a solution

    I have trying to figure out from step by step, however, I do not find what is wrong or what the error is. Could someone help with this problem? Thanks!
     
  2. jcsd
  3. Jan 20, 2013 #2

    haruspex

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    Hint: Is it necessarily the same C in every case?
    Btw, I don't believe integral of dx/x = ln(|x|) + C in the first place. Shouldn't it be (x/|x|)ln(|x|) + C (i.e. the sign reverses for x < 0)?
     
  4. Jan 20, 2013 #3

    LCKurtz

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    Everything you have written is correct, including the ##\neq## sign in the last one. Why would you expect them to be equal when you stuck an extra 3 on the right side?
     
  5. Jan 20, 2013 #4

    haruspex

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    I think the point is that if you take the 1/3 outside then do the integral you get ln(x)/3, but if you leave the 1/3 inside and use the chain rule you get ln(3x)/3. It appears a paradox, but there's a simple explanation.
     
  6. Jan 20, 2013 #5

    LCKurtz

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    No, it is correct. Take the derivative of ln|x| for the case x < 0 so |x| = -x and you will see it.
     
  7. Jan 20, 2013 #6
    WOW THANKS SO MUCH IT MAKES SUCH A GOOD SENSE.. I do not know why I did not figure it out.
     
  8. Jan 20, 2013 #7
    Wait, but my teacher said the same thing as you did: the hint is is the constant always the same?

    HOw is this realated to the answer you have given?
     
  9. Jan 20, 2013 #8

    LCKurtz

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    Do you understand you are using a different C in your last two equations?
     
  10. Jan 20, 2013 #9
    I actually don't get it.
     
  11. Jan 20, 2013 #10

    haruspex

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    Ah yes, thanks.
     
  12. Jan 20, 2013 #11
    How did you get ln(3x) / 3, by using the chain rule?

    I tried to do it myself, and I am getting a different result as I thought
     
  13. Jan 20, 2013 #12

    LCKurtz

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    Forget the absolute values and the C for a second. For x > 0, one answer is (1/3)ln(x) and the other is (1/3)ln(3x). These are not the same because$$
    (1/3)\ln(3x) = 1/3(\ln 3 +\ln x) = (1/3)\ln 3 + (1/3)\ln(x)$$Your two basic answers differ by a constant, which can be included in the constant of integration. So both answers are correct but the constants of integration are different.
     
  14. Jan 20, 2013 #13
    Wow that just blew my mind.. Thank you so much!
     
  15. Jan 21, 2013 #14

    SteamKing

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    Don't forget that (1/3) ln (x) also is equal to ln (x^(1/3))
     
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