How to Calculate the Integral of e^{1/x}dx?

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Discussion Overview

The discussion centers around the calculation of the integral of e^{1/x}dx, exploring its properties and potential methods for evaluation. The scope includes theoretical aspects of integration and the nature of the integral itself.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • Some participants note that the integral does not have an elementary answer and suggest using integral tables or computational tools to find the area under the curve.
  • One participant mentions that using Maple yields an answer in terms of the exponential integral function \mathrm{Ei}_1(x).
  • Another participant proposes obtaining the Taylor expansion of the function and integrating that as a possible approach.
  • A later reply questions the context of the integral and suggests that it is straightforward to use Liouville's theorem to demonstrate that the indefinite integral of e^{1/x} is non-elementary.

Areas of Agreement / Disagreement

Participants generally agree that the integral does not have an elementary form, but multiple approaches and interpretations regarding its evaluation and context remain. The discussion is unresolved regarding the best method to approach the integral.

Contextual Notes

Some limitations include the dependence on definitions of elementary functions and the unresolved nature of the integral's evaluation methods.

pamoriano
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Hello everyone,

How do I to figure out the integral of e^{1/x}dx.

Thanks in advance,
 
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Unfortunately, this does not have an elementary answer. However, there should be integral tables and tools available that will tell you the area under the curve.
 
If you ask Maple, you get an answer in terms of [tex]\mathrm{Ei}_1(x)[/tex]
 
Get the Taylor expansion and integrate that.
 
pamoriano said:
Hello everyone,

How do I to figure out the integral of e^{1/x}dx.

Thanks in advance,
What is the context of this integral? Why do you need to figure it out?

(It is straightforward to use Liouville's thereom to prove that the indefinite integral of e^{1/x} is non-elementary, that is, cannot be expressed in finite terms of elementary functions).
 

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