- #1
eyesontheball1
- 31
- 0
Evaluate the following integral:
\int_0^{∞} \frac{e^{-(x+x^{-1})}}{x}dx
\int_0^{∞} \frac{e^{-(x+x^{-1})}}{x}dx
Last edited:
How to Evaluate the Integral \(\int_0^{∞} \frac{e^{-(x+x^{-1})}}{x}dx\)?
- Context: Graduate
- Thread starter eyesontheball1
- Start date
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- Tags
- Integral
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Discussion Overview
The discussion revolves around evaluating the integral \(\int_0^{∞} \frac{e^{-(x+x^{-1})}}{x}dx\), with participants exploring different methods for both symbolic and numerical evaluation. The scope includes mathematical reasoning and potential connections to special functions.
Discussion Character
- Exploratory, Mathematical reasoning, Debate/contested
Main Points Raised
- Some participants propose that the integral can be evaluated numerically, with one suggesting a specific numerical value of .227788.
- Others clarify that the integral should be evaluated symbolically rather than numerically.
- A participant suggests expressing \(e\) in terms of a limit involving \(n\) and considers a substitution \(u = \frac{1}{x}\) to facilitate evaluation.
- Another participant argues that the integral does not converge, citing the antiderivative involving the exponential integral function and indicating issues at the lower limit of integration.
- One participant mentions that the integral can be expressed in terms of a Bessel function, providing an attachment for reference.
Areas of Agreement / Disagreement
Participants express differing views on the convergence of the integral, with some asserting it diverges while others explore methods of evaluation. The discussion remains unresolved regarding the integral's convergence and the validity of proposed methods.
Contextual Notes
Limitations include potential misunderstandings about the convergence of the integral and the definitions of functions involved, as well as the need for clarity in the evaluation methods proposed.
- #2
joeblow
- 71
- 0
.227788.
(?)
- #3
eyesontheball1
- 31
- 0
Evaluate the following integral (symbolically and not numerically), should've specified that.
- #4
joeblow
- 71
- 0
My guess is that you express e = (1+1/x)^x then work with that.
- #5
Whovian
- 651
- 3
joeblow said:
Since x is already a variable in the problem, I assume you mean ##e=\lim\limits_{n\to0}\left(\left(1+\dfrac1n \right)^n\right)##?
For some reason, I feel like some sort of substitution of ... wait a second ...
How about the substitution ##u=\dfrac1x##?
- #6
Millennial
- 296
- 0
The integral given does not converge. Its antiderivative is \displaystyle e^{-1/x}\text{Ei}(-x) where Ei is the exponential integral function. I think simply plugging in zero for x shows why it wouldn't converge.
- #7
- #8
eyesontheball1
- 31
- 0
Thank you again JJacquelin! I can always count on you! :)
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