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