Extracting a constant variable from integral

In summary, during a conversation about a simple math problem in classical physics, the speaker mentioned that a surplus of a constant variable "b" was extracted from an integral using manipulation. It was then discussed that the extra constant b can come from the d(x/b) and that we can change the variable being integrated with respect to using substitution. This is a useful trick for integrals.
  • #1
shanepitts
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Sorry for the simplicity of the question but I encountered a simple math problem whilst autodidacting myself on classical physics. I seen that a surplus of a specific constant variable "b" was extracted from an integral after a bit of manipulation.

∫xdx was turned into b2∫(x/b)d(x/b)

Does the extra constant b come from the d(x/b)? If so, or if not, how can one change what the integral is being integrated with respect to?

This question is a not a homerwork question.

Thank You
 
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  • #2
If b is a constant, then yes that is valid. We can change the variable you are integrating with respect to, with substitution, and it's a nifty trick for integrals.

d(x/b) is jus dx/b or (1/b) dx so you can take the constant term out like that.
 
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  • #3
Paul Eccles said:
If b is a constant, then yes that is valid. We can change the variable you are integrating with respect to, with substitution, and it's a nifty trick for integrals.

d(x/b) is jus dx/b or (1/b) dx so you can take the constant term out like that.
Awesome, thanks a bunch
 

Related to Extracting a constant variable from integral

1. How do you extract a constant variable from an integral?

To extract a constant variable from an integral, you can use the constant multiple rule or the power rule. The constant multiple rule states that the integral of a constant times a function is equal to the constant times the integral of the function. The power rule states that the integral of x^n is equal to (x^n+1)/(n+1) + C, where C is a constant. These rules allow you to extract the constant variable from the integral and solve for the remaining variable.

2. What is a constant variable in an integral?

A constant variable in an integral is a variable that does not change throughout the integration process. It is usually represented by the letter C and is added to the solution as a reminder that there may be multiple solutions or to account for any unknown constant values.

3. Can you remove the constant variable from an integral?

No, you cannot remove the constant variable from an integral. The constant variable is an important part of the solution and cannot be removed. It represents the indefinite nature of the integral and accounts for any unknown constant values that may be present in the problem.

4. Why is it important to extract the constant variable from an integral?

Extracting the constant variable from an integral is important because it allows you to solve for the remaining variable and find the definite solution to the problem. Without extracting the constant variable, the solution would not be complete and may not accurately represent the problem.

5. Are there any exceptions to extracting a constant variable from an integral?

Yes, there are some exceptions to extracting a constant variable from an integral. One exception is when the integral is a definite integral with specific limits of integration. In this case, the constant variable can be omitted as the definite limits provide a specific solution. Another exception is when the integral is a complex or improper integral, in which case the constant variable may not be able to be extracted and the solution may require further manipulation to account for the constant variable.

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