Integral Help exp (-x)/x with 0 to t limits

  • Thread starter piyush
  • Start date
  • #1
6
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I am trying to get this integral but no clues about how to proceed
[tex] \int exp(-x)/x dx [/tex]

with limits from 0 to t ( any real number, non infinity)

I know of the limits are from t to infinity, there is a standard integral known as Ei(x). please help/share ideas in solving this!


Regards and Thanks

Piyush
 

Answers and Replies

  • #2
56
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Mathematica doesn't like it, it tosses Ei[-x] back at me, whatever that is. Not sure if there's an analytical solution.
 
  • #3
Zurtex
Science Advisor
Homework Helper
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I'm not sure it is valid at x=0. If we looks at the infinite series:

[tex]\frac{e^{-x}}{x} = \frac{1}{x} - \sum_{n=0}^{\infty} (-1)^n\frac{x^n}{(n+1)!}[/tex]

Therefore:

[tex]\int \frac{e^{-x}}{x} dx = \ln x - \sum_{n=0}^{\infty} (-1)^{n}\frac{x^{n+1}}{(n+1)!*(n+1)} + C[/tex]

This is clearly not valid for x = 0, which kind of makes sense as [itex]e^{-x}/x[/itex] is not valid at x = 0.
 
  • #4
Gokul43201
Staff Emeritus
Science Advisor
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Since 1/x is non-integrable in any interval that includes 0 (and exp(0)=1) since the integral diverges, so would be (1/x)*exp(-x).

ie. the integral diverges to infinity.
 
  • #5
6
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Thanks guys: I have a few questions on this:

1. If I integrate within limits say 0.001 to t, is it going to introduce a lot of error in subsequent calculations that i carry out?

2. Actually I am trying to obtain this integral


[tex] \int f(r,x)dx [/tex] with limits 0 to t
where f(r,x) is the Laplacian inverse of F(r, s) where F(r,s) is

[tex] \frac{K_{o}(r\sqrt{s})}{\sqrt{s}K_{1}(\sqrt{s})} [/tex]

Ko and K1 are modified Bessel functions of zero and first order
Some simplifications resulted in the integral that I had put initially..

Can anyone please suggest a way out?
 
  • #6
394
2
If we know t>0 then what is the best way to integrate this problem, without computer help?
 

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