Solving Int. e^(-x^2) w/ Change of Variable: Help Needed

In summary, the conversation discusses the proof of the integral \int_{0}^{+\infty}e^{-x^2}dx = \frac{\sqrt{\pi}}{2} and the use of a change of variable to justify the equality. The conversation also mentions the use of Fubini's theorem and polar substitution, but the book only deals with functions of one real variable. The summary concludes with the explanation of how \sqrt{n}x = y leads to the equality in question.
  • #1
Castilla
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0
I am trying to follow a proof of
[tex] \int_{0}^{+\infty}e^{-x^2}dx = \frac{\sqrt{\pi}}{2} [/tex] but the first impasse I find is that, "with the change of variable" [tex]\sqrt{n}x = y [/tex] they justify this equality:
[tex] {\frac{1}{\sqrt{n}}\int_{0}^{+\infty}e^{-y^2}dy = \int_{0}^{+\infty}e^{-nx^2}dx [/tex].

Maybe you can help me to see how they did it? Thanks.
 
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  • #2
[tex]\int_{0}^{+\infty}e^{-x^2}dx[/tex]
is just the half of
[tex]\int_{-\infty}^{+\infty}e^{-x^2}dx[/tex]

The second is usually done with Fubini's theorem and polar substitution.
 
  • #3
Er... but the book from where I take the problem only deals with functions of one real variable... they don't use Fubini...
 
  • #4
[tex]\sqrt{n}x = y[/tex], so, you have: [tex]d(\sqrt{n}x) = \sqrt{n}dx = dy[/tex]
Then, you have:
[tex]\frac{1}{\sqrt{n}} \int_{0} ^ {+ \infty} e ^ {-y^2} dy = \frac{1}{\sqrt{n}} \int_{0} ^ {+\infty} e ^ {-nx ^ 2} (\sqrt{n} dx)[/tex]
[tex]= \sqrt{n} \times \frac{1}{\sqrt{n}} \int_{0} ^ {+\infty} e ^ {-nx ^ 2} dx = \int_{0} ^ {+\infty} e ^ {-nx ^ 2} dx[/tex].
Can you get it now?
Viet Dao,
 
  • #5
Yes, Viet Dao. Really thanks.
 

1. What is the purpose of using a change of variable when solving for the integral of e^(-x^2)?

The purpose of using a change of variable is to simplify the integral and make it easier to solve. In this case, using the change of variable x = u^2 will eliminate the exponent and make the integral easier to integrate.

2. Why is e^(-x^2) a challenging function to integrate?

Evaluating the integral of e^(-x^2) is challenging because it does not have an elementary antiderivative. This means that there is no simple algebraic expression that can be used to find the integral. Therefore, a change of variable is necessary to make the integral solvable.

3. How do you choose the appropriate change of variable for solving the integral of e^(-x^2)?

The appropriate change of variable can be chosen by identifying a pattern or a specific form in the original integral. In the case of e^(-x^2), the form of the integral will suggest the use of a change of variable. Additionally, it is important to choose a change of variable that will simplify the integral and make it easier to solve.

4. What are the steps for solving the integral of e^(-x^2) with a change of variable?

The steps for solving the integral of e^(-x^2) with a change of variable are as follows:

  1. Identify the appropriate change of variable.
  2. Apply the change of variable to the integral.
  3. Simplify the integral using algebraic manipulation.
  4. Use integration techniques such as u-substitution or integration by parts to solve the integral.
  5. Substitute back in the original variable to get the final solution.

5. Are there any other methods for solving the integral of e^(-x^2) besides using a change of variable?

Yes, there are other methods for solving the integral of e^(-x^2) such as using numerical integration techniques like the trapezoidal or Simpson's rule. However, these methods may not provide an exact solution and can be computationally intensive. Another method is to use a power series expansion, but this can also be complicated and time-consuming. Therefore, using a change of variable is often the preferred method for solving this integral.

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