Solve ∫cos^-x/e^x^x? - Get Help Here

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Discussion Overview

The discussion revolves around the integral ∫cos^-x/e^x^x, with participants questioning its formulation and integrability. The scope includes mathematical reasoning and exploration of integrals, particularly focusing on whether the integral can be expressed in terms of elementary functions.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the clarity of the original formula, suggesting it is poorly formed and thus cannot be integrated.
  • Another participant seeks clarification on the variable of integration and the meaning of the expressions involved, specifically regarding the notation used for cosine and the exponential function.
  • Some participants assert that while the function is continuous and integrable, it does not have an antiderivative expressible in elementary functions.
  • There are claims that an antiderivative exists but cannot be neatly expressed, and it may be represented as an infinite series.
  • Concerns are raised about the implications of branch points in the function, which could affect the convergence of series representations.
  • Several participants express disappointment over the lack of a simple antiderivative, with some suggesting that numerical methods may be the only feasible approach for evaluation.
  • One participant emphasizes the need for the original poster to clarify what they mean by "finding" the integral, as this could influence the discussion on integrability.

Areas of Agreement / Disagreement

Participants generally disagree on the clarity of the original expression and whether it can be integrated in a straightforward manner. There is no consensus on the existence of a simple antiderivative, with some asserting it exists in a more complex form while others maintain it cannot be expressed in elementary terms.

Contextual Notes

Participants highlight limitations in the original formulation and the potential for misunderstanding due to notation. The discussion also touches on the complexity of integrals involving functions with branch points, which may affect convergence.

Superposed_Cat
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∫cos^-x/e^x^x? can this be done? A thanks to anyone who can do this.
 
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What you posted makes *no* sense. It's a poorly formed formula. So of course it can't be done.
 
What you wrote doesn't make sense. Cosine to the power of what? I'm assuming the variable of integration is x? What does e^x^x mean: ##{e^x}^x## or ##(e^x)^x##?
And what does "can this be done" mean? Are you asking if it's integrable? Are you asking if it has an elementary derivative?
 
x is variable of integration. its cos to the power of -x. and to pwsnafu. the former.

sorry

pwsnafu said:
And what does "can this be done" mean? Are you asking if it's integrable? Are you asking if it has an elementary derivative?
yes.
 
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Cosine of what to the power of -x, Superposed_Cat?Suppose you do come up with a well-formed formula that bears some resemblance to what you wrote in the original post. Given the mess in the OP, it's almost assured not going to be integrable in terms of the elementary functions.

Example: What is ∫e-x-2dx ? What about ∫sin(t)/t dt? These are of a much simpler form than what you wrote, and yet one cannot express these indefinite integrals as a closed form expression in the elementary functions.
 
oh damn! I am forgetting so many things deepest apologies:redface:
 

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Who cares?
It is a continuous function, and therefore integrable (wherever the numerator is well defined).
It does not have an antiderivative expressible in elementary functions, but over any finite interval, the value of the integral can readily be calculated by numerical means.
 
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Damn, I hoping there was an antiderivative.
 
Superposed_Cat said:
Damn, I hoping there was an antiderivative.
It sure exists an antiderivative. It's just that is impossible to find a neat formula for it.
 
  • #10
arildno said:
It sure exists an antiderivative. It's just that is impossible to find a neat formula for it.

It all depends on what the OP is happy with, of course. Maybe he's happy by just being able to find the graph and a numerical way of computing things. After all, we don't really know more than that about logarithms or sines and cosines either. But still we interpret those functions as well-known.

On the other hand, if you want to express it as known elementary functions, then such a thing is probably impossible. But an antiderivative is certainly there.

So, it is up to the OP to clarify what he means with "finding" the integral.
 
  • #11
Superposed_Cat said:
Damn, I hoping there was an antiderivative.

If f(x) is an integrable function, then ##g(x) = \int_{a}^{x} f(t) \, dt## is an antiderivative. This is why I asked what you mean by "can be done".
 
  • #12
pwsnafu said:
If f(x) is an integrable function, then ##g(x) = \int_{a}^{x} f(t) \, dt## is an antiderivative. This is why I asked what you mean by "can be done".
I strenuously oppose that that is a NEAT formula.
It is a voracious dragon in a mouse's clothing.
:smile:
 
  • #13
Superposed_Cat said:
Damn, I hoping there was an antiderivative.
Of course there's an antiderivative. The problem is that you can't express it in terms of a finite number of operations using only the elementary functions. You can express it, for example, as some kind of infinite series. Good luck developing that, though.

Another issue: that your function involves xx (exx) and cos-x(x) means there's a branch point at x=0. This is going to make the series have a finite radius of convergence. But an antiderivative certainly does exist.
 
  • #14
Superposed_Cat said:
Damn, I hoping there was an antiderivative.

Translation:Damn, I was hoping that there was an antiderivative that could be found non-numerically in less than an hour.
 
  • #15
The answer then is no.

Why would you think anything there is?
 
  • #16
Superposed_Cat said:
Translation:Damn, I was hoping that there was an antiderivative that could be found non-numerically in less than an hour.
The answer to that would be "no" for almost every integrable function.
 
  • #17
As HallsofIvy implies, what you learn in your studies amounts to practically..Nothing.
:smile:
 

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