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If a=b then integral(a) = integral(b) ... 1/2lnx =/= 1/2ln(2x) ?

  1. Feb 14, 2016 #1
    Am I missing something?

    if a = b then

    Integral a = Integral b

    a = dx/2x and b = dx/2x

    a = (1/2) (dx/x) =
    b = [dx/(2x)]

    So far so good...

    Integral of a .. let U = x, du = dx

    Integral of a = (1/2) ln|x| + C

    Integral of b... let U = 2x, du = 2 dx (multiple by (1/2) to balance out numerator only being 1)
    (1/2) Integral (du/u)

    Integral of b = (1/2) ln|2x| + C

    But wait... (1/2) ln|x| =/= (1/2) ln|2x|

    So did I mess something up or is integral (a) not always = to integral (b) given a = b.
    Last edited: Feb 14, 2016
  2. jcsd
  3. Feb 14, 2016 #2


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    You scammed by hiding the difference in the constant. Your C's aren't equal.
  4. Feb 14, 2016 #3


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    du = 2 dx
  5. Feb 14, 2016 #4
    Help me understand why they aren't equal. Surely there isn't a difference between using one integration technique over the other?

    Yes.. updated... but unfortunately that does not change or resolve the issue here or explain any confusion.
    Last edited: Feb 14, 2016
  6. Feb 14, 2016 #5


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    Sure there is. The boundaries change. If you ignore them by hiding them in the C you could add any constant value, e.g. ##-\frac{1}{2}ln2##.
  7. Feb 14, 2016 #6
    Yea that's right. I just evaluated the definite integral from 1 to 2 and they were equal then. I suppose my real confusion is coming from some where else in my (parent) equation (not this one) I'll have to get back to you in a bit.
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