The integration of e^(x^2)

  • Thread starter Thread starter Owais Iftikhar
  • Start date Start date
  • Tags Tags
    integration
Click For Summary
SUMMARY

The integral of e^(x^2) does not have an elementary anti-derivative, as established in calculus literature. The discussion highlights that while e^(-x^2) is associated with the error function (erf), no similar function exists for e^(x^2). Various methods, including the use of polar coordinates and series expansions, are suggested for approximating the integral, but they do not yield a simple closed form. The conversation emphasizes the limitations of traditional integration techniques when applied to non-elementary functions.

PREREQUISITES
  • Understanding of calculus concepts, particularly integration and anti-derivatives.
  • Familiarity with the error function (erf) and its applications in statistics.
  • Knowledge of series expansions and Taylor series.
  • Basic principles of Laplace transforms and their use in integration.
NEXT STEPS
  • Research the properties and applications of the error function (erf).
  • Learn about the Gaussian integral and its significance in probability and statistics.
  • Explore series expansion techniques for approximating integrals of non-elementary functions.
  • Study the principles of Laplace transforms and their role in solving differential equations.
USEFUL FOR

Mathematicians, calculus students, and anyone interested in advanced integration techniques and the properties of non-elementary functions.

  • #31
HallsofIvy said:
I don't know how many times we have to say this- it's given in any good calculus book. The anti-derivative of e^{x^2} is not an "elementary" function.
My best guess, this number is aleph-naught: infinite, but countable.

I have assumed an infinite lifetime for the forum. :wink:
 
Physics news on Phys.org
  • #32
Redbelly98 said:
I have assumed an infinite lifetime for the forum. :wink:

lol :biggrin:

a nice point of view :wink:
 
  • #33
thepatient said:
No.


If u = x^2. du = 2xdx, not 2dx.

You're right. What a stupid error I made.

u = x^2, du = 2x
multiply du * e^u = 2x e^x^2.
 
  • #34
LTA85 said:
You're right. What a stupid error I made.

u = x^2, du = 2x
multiply du * e^u = 2x e^x^2.
\intex2dx If:

u = x2
du = 2xdx

You can't substitute dx by du, since dx = du/(2x), not just du. You would have dx in terms of two variables, you can't integrate x in terms of u. The anti-derivative of apples can't be oranges.In no nice way is this function integrable.
 

Similar threads

Undergrad Gaussian integral by differentiating under the integral sign
  • · Replies 19 ·
Replies
19
Views
4K
High School Is this identity containing the Gaussian Integral of any use?
  • · Replies 8 ·
Replies
8
Views
2K
High School Integration by Parts, an introduction I get confused with
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Multiplying divergent integrals using Hardy fields approach
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Different Substitution for Solving an Integral
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Find f(z) given f(x, y) = u(x, y) + iv(x,y)
  • · Replies 8 ·
Replies
8
Views
845
High School Calculating shaded area: I'm getting discrepancies between methods
  • · Replies 3 ·
Replies
3
Views
3K
Graduate How do I apply the method of steepest descents to this integral?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Did Math DF's integral calculator make a glaring mistake?
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad ##L^2## square integrable function Hilbert space
  • · Replies 11 ·
Replies
11
Views
3K