Can erf(x) be used to solve e^(x^2)?

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SUMMARY

The discussion centers on the integral of the function e^(x^2) and its relationship with the Error Function, erf(x). It is established that while e^(x^2) is continuous and possesses an integral, its anti-derivative is not an elementary function. The Error Function, erf(x), is defined as the anti-derivative of e^(-x^2), highlighting the distinction between these two functions. Therefore, erf(x) cannot be directly used to solve the integral of e^(x^2).

PREREQUISITES
  • Understanding of continuous functions and their properties
  • Familiarity with anti-derivatives and elementary functions
  • Knowledge of the Error Function, erf(x)
  • Basic calculus concepts, particularly integration
NEXT STEPS
  • Research the properties of the Error Function, erf(x)
  • Explore the concept of non-elementary functions in calculus
  • Learn about integration techniques for functions without elementary anti-derivatives
  • Study the applications of the Error Function in probability and statistics
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Mathematicians, calculus students, and anyone interested in advanced integration techniques and the properties of special functions like the Error Function.

newton1
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does the integrate e^(x^2) can solve??
i think is no...
but why??
 
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That depends upon exactly what you mean.

Since e^(x^2) is a continuous function, yes, it HAS an integral (anti-derivative). Every continuous function (and many non-continuous functions) is the derivative of some function and therefore has an anti-derivative.

Is that anti-derivative any "elementary function" (defined as polynomials, rational functions, exponentials, logarithms, trig functions and combinations of them)? No, if fact for most functions the anti-derivative is not an elementary function. (There are more functions in heaven and Earth than are dreamed of in your philosophy, Horatio!)

Of course one can always DEFINE a new function to do the job. I don't know specifically about e^(x^2) but the ERROR FUNCTION, Erf(x) is defined as an anti-derivative of e^(-x^2).
 
eh...

may i ask what is Error Function??
 

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