# Evaluate the following integral

1. Mar 1, 2013

### eyesontheball1

Evaluate the following integral:

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

Last edited: Mar 1, 2013
2. Mar 1, 2013

### joeblow

Re: Challenge

.227788.

(?)

3. Mar 1, 2013

### eyesontheball1

Re: Challenge

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

4. Mar 1, 2013

### joeblow

Re: Challenge

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

5. Mar 1, 2013

### Whovian

Re: Challenge

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. Mar 2, 2013

### 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.

7. Mar 2, 2013

### JJacquelin

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

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8. Mar 2, 2013

### eyesontheball1

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