The integral you have written is:

$\int e^{-x^2} \, dx$

This is a Gaussian integral, and it doesn't have an elementary closed-form solution (in terms of basic functions like polynomials, exponentials, or trigonometric functions). However, it is commonly represented by a special function known as the error function (denoted as $\text{erf}(x)$).

The integral of $e^{-x^2}$ can be expressed as:

$\int e^{-x^2} \, dx = \frac{\sqrt{\pi}}{2} \, \text{erf}(x) + C$

Where $\text{erf}(x)$ is the error function, and $C$ is the constant of integration.

If you need more details on the error function or how it is used, let me know!

Would you like any further clarification on this integral or the error function?

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Here are some follow-up questions to explore more:

1. What is the definition of the error function ($\text{erf}(x)$)?
2. Can you compute definite integrals involving $e^{-x^2}$?
3. How is the Gaussian integral used in probability theory?
4. What are some numerical methods to approximate integrals like $\int e^{-x^2} \, dx$?
5. What is the history behind the discovery of the error function?

Tip: You can look up tables or use numerical methods to compute values for integrals involving the error function.