The integral of $e^{-x^2}$ does not have a simple closed-form solution in terms of elementary functions. However, it can be expressed using the error function, commonly denoted as $\text{erf}(x)$. Specifically, the indefinite integral of $e^{-x^2}$ is:

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

where $\text{erf}(x)$ is the error function, defined as:

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

For definite integrals, such as $\int_{-\infty}^{\infty} e^{-x^2} \, dx$, the result is well-known and equal to:

$\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}$

Would you like further details or have any specific questions?

Here are 5 related questions:

1. How is the error function $\text{erf}(x)$ derived?
2. What is the Gaussian integral and why is it important?
3. How can $e^{-x^2}$ be approximated numerically?
4. What are the applications of the error function in probability and statistics?
5. Can you explain how $\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}$ is computed?

Tip: The Gaussian integral appears frequently in statistics and quantum mechanics, especially in normal distributions.