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Math Help - Flaw in Integration?

  1. #1
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    Flaw in Integration?

    1/2 times the integral of 1/x is 1/2 lnx

    however when i integrate the same problem as 1/2x and set u to 2x, I get

    1/2 ln2x

    Why are the answers not the same?
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  2. #2
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    Quote Originally Posted by Simon777 View Post
    1/2 times the integral of 1/x is 1/2 lnx

    however when i integrate the same problem as 1/2x and set u to 2x, I get

    1/2 ln2x

    Why are the answers not the same?
    they are the same antiderivatives .

    (1) \frac{1}{2} \ln|x| + C

    (2) \frac{1}{2} \ln|2x| + C = \frac{1}{2} \ln|x| + \ln{2} + C =

    they only differ by a constant.
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  3. #3
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    Quote Originally Posted by skeeter View Post
    they are the same antiderivatives .

    (1) \frac{1}{2} \ln|x| + C

    (2) \frac{1}{2} \ln|2x| + C = \frac{1}{2} \ln|x| + \ln{2} + C =

    they only differ by a constant.
    So the ln2 is just taken out because it can be put in with the constant right?
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  4. #4
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    Quote Originally Posted by Simon777 View Post
    1/2 times the integral of 1/x is 1/2 lnx

    however when i integrate the same problem as 1/2x and set u to 2x, I get

    1/2 ln2x

    Why are the answers not the same?
    Remember that anti derivatives are only unique upto a constant. If you use the FTC you will get the same number out i.e

    \frac{1}{2}\int_{1}^{2}\frac{1}{x}dx=\ln(2)
    Or using the other def you get

    \int_{1}^{2}\frac{1}{2x}dx=\frac{1}{2}\ln(2x)\bigg  |_{1}^{2}=\ln(2x)^{\frac{1}{2}}\bigg|_{1}^{2}=\ln(  \sqrt{4})-\ln(\sqrt{1})=\ln(2)

    they are the same
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  5. #5
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    Quote Originally Posted by TheEmptySet View Post
    \frac{1}{2}\int_{1}^{2}\frac{1}{x}dx=\ln(2)
    Shouldn't that answer be 1/2 ln2 or ln square root of 2?
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