How to Calculate the Integral of e^{1/x}dx?

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In summary, the conversation is about finding the integral of e^{1/x}dx and the response is that there is no elementary answer, but there are tools and methods available to find the area under the curve. One suggestion is to use Maple to get an answer in terms of \mathrm{Ei}_1(x) or to use the Taylor expansion and integrate that. It is also mentioned that Liouville's theorem can be used to prove that the indefinite integral of e^{1/x} is non-elementary.
  • #1
pamoriano
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Hello everyone,

How do I to figure out the integral of e^{1/x}dx.

Thanks in advance,
 
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  • #2
Unfortunately, this does not have an elementary answer. However, there should be integral tables and tools available that will tell you the area under the curve.
 
  • #3
If you ask Maple, you get an answer in terms of [tex]\mathrm{Ei}_1(x)[/tex]
 
  • #4
Get the Taylor expansion and integrate that.
 
  • #5
pamoriano said:
Hello everyone,

How do I to figure out the integral of e^{1/x}dx.

Thanks in advance,
What is the context of this integral? Why do you need to figure it out?

(It is straightforward to use Liouville's thereom to prove that the indefinite integral of e^{1/x} is non-elementary, that is, cannot be expressed in finite terms of elementary functions).
 

1. What is the formula for the integral of e^(1/x)dx?

The formula for the integral of e^(1/x)dx is ∫ e^(1/x)dx = xln(x) + C, where C is a constant.

2. What is the domain of the function e^(1/x)?

The domain of e^(1/x) is all real numbers except x = 0, since the function is undefined at that point.

3. Is it possible to find a closed form solution for the integral of e^(1/x)dx?

No, it is not possible to find a closed form solution for the integral of e^(1/x)dx. It can only be expressed in terms of the logarithmic function.

4. How do you solve the integral of e^(1/x)dx by substitution?

To solve the integral of e^(1/x)dx by substitution, let u = 1/x and du = -1/x^2 dx. Then, the integral becomes ∫ e^u * (-1/u^2) du. Using the formula for integration by substitution, the solution is -e^u + C = -e^(1/x) + C.

5. Can the integral of e^(1/x)dx be evaluated numerically?

Yes, the integral of e^(1/x)dx can be evaluated numerically using numerical integration methods such as the trapezoidal rule or Simpson's rule.

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