1. ## 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?

2. Originally Posted by Simon777
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) $\displaystyle \frac{1}{2} \ln|x| + C$

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

they only differ by a constant.

3. Originally Posted by skeeter
they are the same antiderivatives .

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

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

they only differ by a constant.
So the ln2 is just taken out because it can be put in with the constant right?

4. Originally Posted by Simon777
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

5. Originally Posted by TheEmptySet
$\displaystyle \frac{1}{2}\int_{1}^{2}\frac{1}{x}dx=\ln(2)$
Shouldn't that answer be 1/2 ln2 or ln square root of 2?