# Flaw in Integration?

• Mar 28th 2010, 10:24 AM
Simon777
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?
• Mar 28th 2010, 10:28 AM
skeeter
Quote:

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

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

they only differ by a constant.
• Mar 28th 2010, 10:32 AM
Simon777
Quote:

Originally Posted by skeeter
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?
• Mar 28th 2010, 10:33 AM
TheEmptySet
Quote:

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

$\frac{1}{2}\int_{1}^{2}\frac{1}{x}dx=\ln(2)$
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
• Mar 28th 2010, 12:21 PM
Simon777
Quote:

Originally Posted by TheEmptySet
$\frac{1}{2}\int_{1}^{2}\frac{1}{x}dx=\ln(2)$

Shouldn't that answer be 1/2 ln2 or ln square root of 2?