Integral of derivative inconsistent

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    Derivative Integral
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Discussion Overview

The discussion revolves around the issue of integrating a derivative and not obtaining the original function. Participants explore the implications of constants in integration and the nuances of mathematical expressions.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant presents a specific example where the integral of the derivative does not yield the original function, questioning if a fundamental principle is overlooked.
  • Another participant suggests that the issue arises from an incorrect integral, hinting at the importance of including a constant in the solution.
  • A participant expresses confusion about how a constant could account for the discrepancy in the numerator of the integral result.
  • Further suggestions are made to manipulate the expression obtained from the integral to find a way to reach the original equation.
  • One participant concludes that by recognizing the constant as a specific value, they can reconstruct the original function from the integral result.
  • Concerns about communication style are raised, with one participant feeling that another's tone was condescending, leading to a brief discussion about intent and interpretation.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the initial misunderstanding, but there is agreement on the importance of constants in integration. The discussion includes differing perspectives on communication style.

Contextual Notes

Participants do not fully resolve the mathematical steps involved in the integration process, and there is an assumption that the constant of integration plays a crucial role in the outcome.

dc20
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Hi,
I ran into a situation I haven't experienced before where the integral of the derivative doesn't get me the original equation.
Is there a fundamental principle I am missing with this specific example ?
Derivative of this...
(x^2) / ( (x^2)-4 )
...gives...
(-8*x) / ( (x^2-4)^2 )
But integral of (that)...
(-8*x) / ( (x^2-4)^2 )
...gives...
(4) / ( (x^2-4) )
...instead of (x^2) / ( (x^2)-4 ) (original equation).
 
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That's because you have your integral wrong! Remember that little extra bit that you're supposed to include, but never bother? ...
 
If you're referring to the constant that accompanies an integral answer, I don't see how that gets me an "x" in the numerator. That would just give me

(4) / ( (x^2-4) ) + C

(I'm doing the derivative first)

Another hint please ?
 
Last edited:
(1) Have you tried doing anything at all with the expression (4) / ( (x^2-4) ) + C ?


(2) Try stating mathematically what you would like to happen.


Either of these by itself should lead you to something that would make you happy. Have you tried either yet? These are the sorts of things that you should have already thought of -- get into that habit if you aren't there yet!
 
ok, I see: C = ( (x^2-4) ) / ( (x^2-4) ) = 1

giving in the numerator 4 + (x^2-4) = x^2 , resulting in the original equation (x^2) / ( (x^2)-4 ).


thanks, but I think your attitude or at least the way you come across in text could use an adjustment. The condescending comments aren't needed. Your comment was edited here, but came thru in the email and I can tell you I am far from lazy.
 
I tend to have a rather silly attitude -- I edited it once I realized that it would be interpreted as condascending (which I had not intended) instead of general silliness. Sorry if it bothered you.
 
no problem. thanks again for your help.
 

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