Antidifferentiating To \int^{2c}_{c} e^f(x)dx = 7

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Discussion Overview

The discussion revolves around the evaluation of the integral \(\int^{2c}_{c} e^{f(x)}dx = 7\) and the subsequent question of how to compute \(\int^{2}_{1} e^{f(cx)}dx\). Participants explore the validity of different methods for finding antiderivatives and applying the Fundamental Theorem of Calculus (FTC) in this context.

Discussion Character

  • Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant questions the validity of finding an antiderivative for \(\int^{2c}_{c} e^{f(x)}dx\) and applying the FTC to evaluate \(\int^{2}_{1} e^{f(cx)}dx\), suggesting that their teacher indicated an assumption about \(c\) being equal to 1 was incorrect.
  • Another participant proposes that the antiderivative of \(e^{f(x)}\) could be expressed as \(\frac{e^{f(x)}}{f'(x)} + C\) and questions whether this approach is valid in general.
  • A third participant challenges the idea of finding an antiderivative without knowing the function \(f\), stating that the proposed antiderivative form is incorrect and suggesting that substitution is a more straightforward method.
  • A later reply expresses realization and gratitude for the clarification provided by other participants.

Areas of Agreement / Disagreement

Participants express differing views on the appropriateness of using antiderivatives in this context, with some advocating for substitution as the preferred method. No consensus is reached regarding the validity of the proposed antiderivative forms or the assumptions made about \(c\).

Contextual Notes

Participants highlight the uncertainty surrounding the function \(f\) and the implications of assuming specific values for \(c\). There are unresolved questions about the applicability of the FTC in this scenario.

snipez90
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If [tex]\int^{2c}_{c}[/tex] e^f(x)dx = 7, then what is [tex]\int^{2}_{1}[/tex] e^f(cx)dx?

Ok I understood that u-sub is the easiest way to solve this, but what is wrong with trying to find an antiderivative of the first expression and the second, using the FTC and substituting to get the answer? I did it that way and the teacher said I was making the assumption that C = 1 (i think 0 or 1) but he didn't explain to me why it was wrong.
 
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Is it wrong to say that the antiderivative of e^f(x) is e^f(x)/f'(x) + C and then applying FTC? Or does that only work for one case? I don't know if such manipulations are permissible. Just wondering if my "brute-forcing" is even correct.
 
You don't know what f is, so how do you expect to find an antiderivative? And no e^f(x)/f'(x)+C is not correct, to see why what happens when you differentiate it? I think a substitution is definitely the easiest way, and off the top of my head I can't think of a different way to do it.
 
totally see it now, man i was blind. thank you.
 

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