My new U substitution approach? is this legal?

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Discussion Overview

The discussion revolves around a proposed method for performing integration using higher derivatives in the context of u-substitution, specifically the "Mancini conjecture." Participants explore the implications of differentiating the substitution variable multiple times and how this might affect the integration process.

Discussion Character

  • Exploratory, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant introduces the "Mancini conjecture," suggesting that repeatedly differentiating the substitution variable could simplify integration by eventually yielding a constant.
  • The same participant questions whether taking higher derivatives of the substitution variable could allow for a new approach to integration, proposing that compensations might be necessary.
  • Another participant argues that only the first derivative is relevant for variable substitution in integration, implying that higher derivatives do not contribute meaningfully.
  • A different participant expresses skepticism about the validity of taking a second indefinite integral, noting the necessity of adding a constant during integration.
  • One participant encourages experimentation with the proposed method, suggesting that checking results through differentiation could clarify the validity of the approach.

Areas of Agreement / Disagreement

Participants do not reach a consensus. There are competing views on the validity and utility of using higher derivatives in the context of u-substitution for integration.

Contextual Notes

The discussion highlights limitations in the proposed method, particularly the reliance on higher derivatives and the need for compensatory adjustments during integration, which remain unresolved.

mancini0
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Allow me to explain my new theory, The "Mancini conjecture."

Ok...lets say I have an integral like (4-x^2)^(1/2) dx.

and letting u = 4-x^2, we get du/dx = -2x,

and if I took the second derivative of du/dx...i would get -2

this would be ideal, because I would then have du'' = -2 dx, or -1/2 du'' = dx.

So what I'm trying to ask is, if a u substitution doesn't work out clean...that is, when i take du/dx and I end up with something that doesn't help me, what exactly is stopping me from going on, that is keep taking the next derivative...just keep taking the nth derivative until eventually I end up with a constant, which I can then pull out in front of the integral. Of course, if I do this, I would have to compensate for this somehow. I am having difficulty seeing how I would compensate for this, but I have a feeling like there is an obvious compensation that could be made. Maybe take the antiderivative of the answer you get using du'' instead of du. Could I just add an another integral sign each time i differentiate u another time? that is take the antiderivative of an antiderivative as many times as i differentiate u? Maybe I would have to differentiate u again each time I take the next derivative of du/dx . Is the integral of u du the same as the integral of u' du''?

I have this feeling like I'm on to something, because if the exponent is positive, wouldn't I eventually get a constant after n differentiations? Wouldn't this make taking integrals super easy? Just differentiate n times until you get a constant, apply n compensations to the integral for doing that, and pull the constant out front. So once I figure out how to relate the integral of u du'' to u du, I will claim my nobel prize. My intuition is "if you take the derivative of du n times, just take the nth antiderivative of your answer." I just thought of it today doing my calculus 2 homework, and I became excited. We haven't learned integration by parts yet. I should sit down with a pencil and paper and test my intuition, but I'm a B student
and i suck at integrals, and USA mens hockey is on soon. Let me know if I'm on to something. Thanks!
 
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I can't quite figure out what you are trying to do. However when changing variables for integration purposes, only the first derivative makes sense. You are replacing dx by (dx/du)du. Higher derivatives are irrelevant.
 
I don't believe that would work. I mean you couldn't really take a second indefinite integral because you must add a constant when integrating.
 
Try it and see... You can easily check the result of an integration problem by differentiating the answer to see if you get the original integrand.

In this case, as noted in another reply, it doesn't work.
 

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