Is One or Both of the Integral Solutions Correct?

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In summary, there are two ways to solve the indefinite integral \int(x^2+6)(2x)dx: by multiplying the terms and using the sum rule, yielding \frac{x^4+12x^2}{2} + C, or by substitution, yielding \frac{(x^2+6)^2}{2} + C. Both solutions are equally correct, but the second one, using the chain rule, may be preferred for its neatness and ease of checking. The constants C and C1 are just arbitrary and can be combined, making the integral look cleaner in some cases.
  • #1
DocZaius
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Consider this indefinite integral:

[tex]\int(x^2+6)(2x)dx[/tex]

There are two ways I could approach solving it. The first one would be to multiply the terms, then solve using the sum rule. That approach would yield this solution:

[tex]\frac{x^4+12x^2}{2} + C[/tex]

The other way would be by substitution. It would yield this:

[tex]\frac{(x^2+6)^2}{2} + C[/tex]

The second constant is 18 less than the first one. I realize that since they're arbitrary constants, it might not matter, but I want to be clear. My question is:

Strictly speaking, are both solutions equally correct or is only one correct? If it's neither completely equal or one being the only correct answer, and it's merely a matter of preference, could you elaborate in what context which solution is preferable?

Thanks!
 
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  • #2
Both are equal since C is just an arbitrary constant, although one might be more "correct" than the other depending on what method of integration you are supposed to do and how the result might be expected to look like. In some problems where you have a number and the constant of integration, you can combine the two. For example, you might have 18 + C1, then let C = 18 + C1, so you have just C for the constant, which makes some integrals look cleaner.
 
  • #3
Yes, both solutions are correct,

but the second (chain rule) one is neater, since you can instantly both do it and check it. :smile:
 

1. What is the purpose of checking for integral solutions in a scientific study?

The purpose of checking for integral solutions in a scientific study is to verify the accuracy and reliability of the results. Integral solutions, or whole number solutions, are considered more precise and less prone to error compared to non-integral solutions. Therefore, checking for integral solutions can help ensure the validity of the study's findings.

2. How do you determine if an integral solution is correct?

An integral solution is considered correct if it satisfies all the given conditions and constraints of the problem. This means that the solution must be a whole number and must fulfill all the equations, inequalities, or other criteria set in the problem. Additionally, the integral solution must be logical and make sense in the context of the study.

3. What are the potential consequences of using an incorrect integral solution in a scientific study?

Using an incorrect integral solution in a scientific study can lead to inaccurate or invalid results. This can potentially impact the overall conclusions and implications of the study. It can also undermine the credibility and reliability of the study, especially if the incorrect solution is used as a basis for further research or applications.

4. How do you handle cases where there are multiple integral solutions to a problem?

In cases where there are multiple integral solutions to a problem, the scientist must carefully evaluate each solution and determine which one is the most reasonable and appropriate. This can involve considering factors such as simplicity, consistency with other findings, and practicality. If there are still uncertainties or discrepancies, further analysis and experimentation may be necessary to confirm the correct solution.

5. Are integral solutions always necessary in a scientific study?

No, integral solutions are not always necessary in a scientific study. It depends on the specific research question and methodology. In some cases, non-integral solutions may be more relevant or appropriate. However, it is generally recommended to check for integral solutions as part of the rigorous and systematic approach to scientific inquiry.

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