How to find the error in this particular integration process?

[taeyeong14]

Homework Statement

The problem, more like a riddle, that our teacher gave it to us was to find the error in this particular equation:

integral of 1/x dc = ln (absolute value of (x)) + C ; perfectly fine
integral of dx/3x = 1/3 integral of dx/x ; also fine
1/3 integral dx/x = 1/3 ln (absolute value of x) + C ; not sure from here
integral of dx/3x ≠ 1/3 ln (absolute value of 3x) + C

The Attempt at a Solution

I have trying to figure out from step by step, however, I do not find what is wrong or what the error is. Could someone help with this problem? Thanks!

 

[haruspex]

Hint: Is it necessarily the same C in every case?
Btw, I don't believe integral of dx/x = ln(|x|) + C in the first place. Shouldn't it be (x/|x|)ln(|x|) + C (i.e. the sign reverses for x < 0)?

 

[LCKurtz]

Homework Statement

The problem, more like a riddle, that our teacher gave it to us was to find the error in this particular equation:

integral of 1/x dc = ln (absolute value of (x)) + C ; perfectly fine you mean dx
integral of dx/3x = 1/3 integral of dx/x ; also fine
1/3 integral dx/x = 1/3 ln (absolute value of x) + C ; not sure from here
integral of dx/3x ≠ 1/3 ln (absolute value of 3x) + C

The Attempt at a Solution

I have trying to figure out from step by step, however, I do not find what is wrong or what the error is. Could someone help with this problem? Thanks!

Everything you have written is correct, including the $\neq$ sign in the last one. Why would you expect them to be equal when you stuck an extra 3 on the right side?

 

[haruspex]

Everything you have written is correct, including the $\neq$ sign in the last one. Why would you expect them to be equal when you stuck an extra 3 on the right side?

I think the point is that if you take the 1/3 outside then do the integral you get ln(x)/3, but if you leave the 1/3 inside and use the chain rule you get ln(3x)/3. It appears a paradox, but there's a simple explanation.

 

[LCKurtz]

Hint: Is it necessarily the same C in every case?
Btw, I don't believe integral of dx/x = ln(|x|) + C in the first place. Shouldn't it be (x/|x|)ln(|x|) + C (i.e. the sign reverses for x < 0)?

No, it is correct. Take the derivative of ln|x| for the case x < 0 so |x| = -x and you will see it.

 

[taeyeong14]

I think the point is that if you take the 1/3 outside then do the integral you get ln(x)/3, but if you leave the 1/3 inside and use the chain rule you get ln(3x)/3. It appears a paradox, but there's a simple explanation.

WOW THANKS SO MUCH IT MAKES SUCH A GOOD SENSE.. I do not know why I did not figure it out.

 

[taeyeong14]

I think the point is that if you take the 1/3 outside then do the integral you get ln(x)/3, but if you leave the 1/3 inside and use the chain rule you get ln(3x)/3. It appears a paradox, but there's a simple explanation.

Wait, but my teacher said the same thing as you did: the hint is is the constant always the same?

HOw is this realated to the answer you have given?

 

[LCKurtz]

WOW THANKS SO MUCH IT MAKES SUCH A GOOD SENSE.. I do not know why I did not figure it out.

Do you understand you are using a different C in your last two equations?

 

[taeyeong14]

Do you understand you are using a different C in your last two equations?

I actually don't get it.

 

[haruspex]

No, it is correct. Take the derivative of ln|x| for the case x < 0 so |x| = -x and you will see it.

Ah yes, thanks.

 

[taeyeong14]

Ah yes, thanks.

How did you get ln(3x) / 3, by using the chain rule?

I tried to do it myself, and I am getting a different result as I thought

 

[LCKurtz]

Forget the absolute values and the C for a second. For x > 0, one answer is (1/3)ln(x) and the other is (1/3)ln(3x). These are not the same because$$
(1/3)\ln(3x) = 1/3(\ln 3 +\ln x) = (1/3)\ln 3 + (1/3)\ln(x)$$Your two basic answers differ by a constant, which can be included in the constant of integration. So both answers are correct but the constants of integration are different.

 

[taeyeong14]

Forget the absolute values and the C for a second. For x > 0, one answer is (1/3)ln(x) and the other is (1/3)ln(3x). These are not the same because$$
(1/3)\ln(3x) = 1/3(\ln 3 +\ln x) = (1/3)\ln 3 + (1/3)\ln(x)$$Your two basic answers differ by a constant, which can be included in the constant of integration. So both answers are correct but the constants of integration are different.

Wow that just blew my mind.. Thank you so much!

 

[SteamKing]

Don't forget that (1/3) ln (x) also is equal to ln (x^(1/3))