The integral of $e^{x^2} \, dx$ does not have a simple elementary antiderivative. This is a well-known problem in calculus, as the function $e^{x^2}$ cannot be integrated in terms of elementary functions (polynomials, exponentials, trigonometric functions, etc.).

However, the integral is typically represented using the error function $\text{erf}(x)$, which is defined as:

$\text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt$

Unfortunately, since the exponent in $e^{x^2}$ is positive, this doesn't directly correspond to the error function.

For practical purposes, the integral of $e^{x^2}$ is often expressed in terms of a series expansion or approximated numerically, but it cannot be simplified to elementary functions.

Would you like to explore an approximation or learn more about series expansions for this kind of integral?

5 Related Questions:

1. What are series expansions, and how are they useful in approximating integrals?
2. How do we numerically approximate non-elementary integrals like $e^{x^2}$?
3. What is the error function, and where does it come into play in integrals?
4. How can we handle integrals that do not have elementary antiderivatives?
5. Are there other common functions with no elementary antiderivative?

Tip: When faced with a function that doesn’t have an elementary antiderivative, numerical methods or series expansions are powerful tools for approximation.