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

[eyesontheball1]

Evaluate the following integral:

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

 

[joeblow]

.227788.

(?)

 

[eyesontheball1]

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

 

[joeblow]

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

 

[Whovian]

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 $u=\dfrac1x$?

 

[Millennial]

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.

 

[JJacquelin]

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

 

[eyesontheball1]

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