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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- Mar 28th 2010, 09:24 AMSimon777Flaw 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? - Mar 28th 2010, 09:28 AMskeeter
- Mar 28th 2010, 09:32 AMSimon777
- Mar 28th 2010, 09:33 AMTheEmptySet
Remember that anti derivatives are only unique upto a constant. If you use the FTC you will get the same number out i.e

$\displaystyle \frac{1}{2}\int_{1}^{2}\frac{1}{x}dx=\ln(2)$

Or using the other def you get

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

they are the same - Mar 28th 2010, 11:21 AMSimon777