Integrate exp(-x^2), dx

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Discussion Overview

The discussion revolves around the integral of the function exp(-x^2) and its related forms, including the error function (erf). Participants explore methods for solving this integral, including double integration techniques and the implications of limits of integration. The conversation also touches on a more general case involving the integral of exp(-(x/C)^k) for k>0.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses confusion over the result involving the error function (erf) when attempting to solve the integral of exp(-x^2).
  • Another participant states that there is no analytic form for the Gaussian integral and suggests using erf tables for definite integrals.
  • A participant references a method involving double integration to solve the integral, providing a link to a related discussion and outlining a polar coordinate approach.
  • There is a note that the double integration method is applicable only for certain definite integrals, prompting a question about the limitations of this approach.
  • One participant reflects on their initial experience with the integral and the perceived elegance of the method for specific limits.
  • A new participant introduces a related problem involving the integral of exp(-(x/C)^k) for k>0, indicating difficulty in solving it for values of k greater than 2.

Areas of Agreement / Disagreement

Participants generally agree that the integral of exp(-x^2) does not have an analytic solution and that the error function is relevant. However, there are differing views on the applicability of the double integration method and the specific limits of integration that affect its use. The discussion regarding the integral of exp(-(x/C)^k) introduces additional complexity and remains unresolved.

Contextual Notes

The discussion highlights the dependence on specific limits of integration for the applicability of certain methods, as well as the unresolved nature of the integral involving exp(-(x/C)^k) for k>0.

Who May Find This Useful

This discussion may be useful for students and practitioners in mathematics and physics who are exploring integrals involving exponential functions, particularly in the context of Gaussian integrals and error functions.

Nimrod
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hi all,

i've tried to solve this thing with Derive, but it gave me some vague erf(x) function (error function??). Is there some gosu-mathematician who can help me solve the integral?

\int exp(-x^2) dx



tnx
 
Last edited:
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There is no analytic form to the gaussian integral. You need to look up values in the erf table for definite integrals.
 
ah, that clarifies a lot, thank you.
 
Tom did this using double integration here: https://www.physicsforums.com/showthread.php?t=25798&page=2

Have fun!

Tom Mattson said:
This integral can be done the same way that the integral of exp(-x2) can be done. First, write the integral of x2exp(-x2) from zero to infinity. Then write the integral of y2exp(-y2) from zero to infinity (they're both exactly the same as your integral). Now multiply the integrands together double integrate over x and y. When you convert to polar coordinates, you will get an integral that can be done by parts. Just don't forget to take the square root at the end.

Note:

x2y2=r4sin2(θ)cos2(θ)
x2+y2=r2
dx dy=rdr dθ
 
master_coda said:
Remember, that that only works for solving certain definite integrals.

Why's that?
 
It's the limits of integration that count here. Say you're trying to integrate a hard function, but there's a neat little trick for working out the integral from zero to infinity. That trick probably won't help you if you're integrating from, say, 1 to 5.729.
 
When I first did that integral (the trick way with the nice limits) I thought it was the neatest thing.
 
integrate exp(-(x/C)^k), dx with k>0 and C>0

Hello to you all,

i've tried hard to solve this problem related with the wind resources, but so far like Tom Mattson said in is post, i solved the problem to k=2, but i can't solve it to any k>0!

integrate exp(-(x/C)^k), dx with k>0 and C>0

Is there anyone willing to help me ?

Perinhas.
 

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