Flaw in Integration?

**Simon777** · March 28th 2010, 09:24 AM

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?

**skeeter** · March 28th 2010, 09:28 AM

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.

**Simon777** · March 28th 2010, 09:32 AM

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?

**TheEmptySet** · March 28th 2010, 09:33 AM

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

**Simon777** · March 28th 2010, 11:21 AM

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?

Thread: Flaw in Integration?

LinkBack

Thread Tools

Display

Flaw in Integration?

Similar Math Help Forum Discussions

Flaw in proof

sketch the region of integration and change the order of integration.

Trig substituion: what is the flaw in my logic?

Where is logic flaw?

Tricky integration, possibly Integration Factor

Search Tags