How to Evaluate the Integral \(\int_0^{∞} \frac{e^{-(x+x^{-1})}}{x}dx\)?

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The integral \(\int_0^{∞} \frac{e^{-(x+x^{-1})}}{x}dx\) is discussed with various approaches to evaluate it symbolically. A substitution \(u = \frac{1}{x}\) is suggested, but it's noted that the integral does not converge. The antiderivative involves the exponential integral function, indicating complexities in evaluation. Additionally, it is mentioned that the integral can be expressed in terms of a Bessel function. The discussion highlights the challenges and methods for tackling this integral.
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Evaluate the following integral:

\int_0^{∞} \frac{e^{-(x+x^{-1})}}{x}dx
 
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(?)
 


Evaluate the following integral (symbolically and not numerically), should've specified that.
 


My guess is that you express e = (1+1/x)^x then work with that.
 


joeblow said:
My guess is that you express e = (1+1/x)^x then work with that.

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##?
 
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.
 
Hi !
the integral can be expressed in terms of a Bessel function (attachment)
 

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Thank you again JJacquelin! I can always count on you! :)
 

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