The integral of \( e^{-x^2} \) does not have a simple closed-form expression in terms of elementary functions. This type of integral is commonly encountered in probability theory and statistics and is related to the Gaussian (normal) distribution.

However, the integral can be expressed using the **error function** (\(\text{erf}(x)\)), which is defined as:

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

Therefore, the indefinite integral of \( e^{-x^2} \) is given by:

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

Where \(C\) is the constant of integration.

For the **definite integral** over the entire real line, we have:

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

This result is important in various applications, particularly in normal distribution calculations.

Would you like more details on how the error function is derived or its applications?

Here are five related questions to expand your understanding:

1. What are the properties of the error function \(\text{erf}(x)\)?
2. How is the Gaussian integral used in probability and statistics?
3. Can the integral of \( e^{-x^2} \) be approximated using numerical methods?
4. What is the relationship between the normal distribution and the Gaussian integral?
5. How does the integral of \( e^{-x^2} \) relate to the heat equation in physics?

**Tip:** For integrals that do not have elementary solutions, special functions like \(\text{erf}(x)\) or numerical methods can often provide valuable approximations.

Explore the integral of e^(-x^2) and how it relates to the error function and Gaussian distribution. Learn about the key formulas, including the Gaussian integral, and their applications in probability and statistics. Suitable for students studying advanced calculus or probability theory.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation