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The problem asks to determine the general formula for the integral:

\[
I_n = \int_{-\infty}^{\infty} x^n e^{-\alpha x^2} \, dx
\]

where \( n = 0, 1, 2, 3, 4, \dots \).

This is a Gaussian-type integral and can be evaluated using techniques related to the properties of Gaussian integrals and moments of the normal distribution.

### Key Considerations:
- When \( n \) is odd, the integral will evaluate to zero due to the symmetry of the integrand. This is because \( x^n e^{-\alpha x^2} \) is an odd function, and the integral over a symmetric interval from \( -\infty \) to \( \infty \) of an odd function is zero.
- When \( n \) is even, the integral will be non-zero and related to the Gamma function or factorials, based on powers of the Gaussian distribution.

### General Approach:

#### 1. Case when \( n = 0 \):
The integral becomes:
\[
I_0 = \int_{-\infty}^{\infty} e^{-\alpha x^2} \, dx
\]
This is a standard Gaussian integral, and its value is known:
\[
I_0 = \sqrt{\frac{\pi}{\alpha}}
\]

#### 2. Case when \( n = 2m \) (even):
For even powers \( n = 2m \), the integral is:
\[
I_{2m} = \int_{-\infty}^{\infty} x^{2m} e^{-\alpha x^2} \, dx
\]
This can be related to the Gamma function or factorials. Specifically, for even \( n \), the result can be expressed as:
\[
I_{2m} = \frac{(2m-1)!!}{(2\alpha)^{m}} \sqrt{\frac{\pi}{\alpha}}
\]
where \( (2m-1)!! \) is the double factorial of \( (2m-1) \), which is the product of all odd numbers up to \( 2m-1 \).

### General Formula:
The general formula for the integral is:
\[
I_n = 
\begin{cases} 
0 & \text{if } n \text{ is odd}, \\
\frac{(n-1)!!}{(2\alpha)^{n/2}} \sqrt{\frac{\pi}{\alpha}} & \text{if } n \text{ is even}.
\end{cases}
\]

Would you like further details or clarification?

Here are 5 related questions to expand your understanding:
1. How do Gaussian integrals arise in physics and probability theory?
2. Can you explain why the integral is zero for odd values of \( n \)?
3. What is the double factorial function, and how does it relate to ordinary factorials?
4. How does the Gamma function generalize factorials, and where is it applied in integration?
5. Can you derive the general formula for higher-order moments of a Gaussian distribution?

**Tip:** When dealing with Gaussian integrals involving powers of \( x \), symmetry considerations are crucial to simplifying the problem.

Determine the general formula for integral ∫x^n e^{-αx^2} dx for n = 0, 1, 2, 3, 4, ...

Learn how to determine the general formula for the Gaussian integral ∫x^n e^{-αx^2} dx for even and odd values of n. This solution explores symmetry, double factorials, and the Gamma function. Suitable for calculus and mathematical physics students.

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