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.

Discover how to integrate exp(-x^2) with detailed explanations on the use of the Gaussian integral and error function. Learn about the key formulas, such as \(\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}\) and how \( e^{-x^2} \) relates to the error function. Suitable for university-level students.

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