How to find the error in this particular integration process?

In summary, the conversation was about a problem given by a teacher to find the error in a particular equation involving integrals. The equation in question was integral of dx/3x ≠ 1/3 ln (absolute value of 3x) + C. The error was eventually found to be using a different constant in the two different equations, leading to different answers. The concept of different constants of integration was also discussed in relation to the problem.
  • #1
taeyeong14
9
0

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!
 
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  • #2
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)?
 
  • #3
taeyeong14 said:

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?
 
  • #4
LCKurtz said:
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.
 
  • #5
haruspex said:
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.
 
  • #6
haruspex said:
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.
 
  • #7
haruspex said:
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?
 
  • #8
taeyeong14 said:
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?
 
  • #9
LCKurtz said:
Do you understand you are using a different C in your last two equations?

I actually don't get it.
 
  • #10
LCKurtz said:
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.
 
  • #11
haruspex said:
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
 
  • #12
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.
 
  • #13
LCKurtz said:
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!
 
  • #14
Don't forget that (1/3) ln (x) also is equal to ln (x^(1/3))
 

FAQ: How to find the error in this particular integration process?

1. How do I know if an integration process is accurate?

The best way to determine the accuracy of an integration process is to compare the results to a known solution or a benchmark. This can be done by using analytical methods, numerical methods, or by performing experiments. If the results are close to the known solution or benchmark, then the integration process is likely accurate.

2. What are the common errors in an integration process?

Some common errors in an integration process include round-off error, truncation error, and convergence error. Round-off error occurs when the computer rounds off numbers during calculations, resulting in a loss of precision. Truncation error occurs when approximations are made during the integration process, leading to a difference between the exact and calculated values. Convergence error occurs when the integration process does not converge to the correct solution.

3. How can I identify and fix errors in an integration process?

To identify and fix errors in an integration process, it is important to carefully review the integration algorithm and make sure all steps are correct. It may also be helpful to check for round-off and truncation errors by using smaller step sizes and comparing the results. If convergence error is the issue, adjusting the integration method or step size may help improve accuracy.

4. What is the role of numerical methods in integration processes?

Numerical methods play a crucial role in integration processes by providing algorithms and techniques for approximating the solution of a mathematical problem. These methods are used to break down complex problems into smaller, more manageable parts that can be solved numerically. They are essential in finding the most accurate and efficient solution possible.

5. Can errors in an integration process be completely eliminated?

No, it is not possible to completely eliminate errors in an integration process. However, by carefully selecting integration methods and step sizes, and by using error analysis techniques, the accuracy of the integration process can be greatly improved. It is important to understand the limitations of the integration process and make adjustments accordingly to minimize errors.

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