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The problem asks for the general formula of the integral:

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

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

This is a standard Gaussian integral problem, where the integrand involves a product of a polynomial \( x^n \) and the Gaussian function \( e^{-\alpha x^2} \). The behavior of the integral depends on whether \( n \) is even or odd.

### Solution:

1. **Odd Powers \( n \)**:
   For any odd \( n \), the integral evaluates to zero. This is due to the fact that \( x^n \) is an odd function and \( e^{-\alpha x^2} \) is an even function, so their product is odd. Integrating an odd function over a symmetric interval (from \( -\infty \) to \( \infty \)) gives zero:
   \[
   I_n = 0 \quad \text{for odd } n.
   \]

2. **Even Powers \( n \)**:
   For even powers of \( n \), the integral can be solved using known results for Gaussian integrals. Specifically, when \( n = 2k \) (where \( k \) is a non-negative integer), the integral has the following general form:

   \[
   I_{2k} = \int_{-\infty}^{\infty} x^{2k} e^{-\alpha x^2} \, dx = \frac{(2k-1)!!}{(2\alpha)^{k}} \sqrt{\frac{\pi}{\alpha}},
   \]
   where \( (2k-1)!! \) denotes the double factorial, defined as:
   \[
   (2k-1)!! = (2k-1)(2k-3)(2k-5)\cdots 3 \cdot 1.
   \]

   For example:
   - For \( n = 0 \) (i.e., \( k = 0 \)): 
     \[
     I_0 = \int_{-\infty}^{\infty} e^{-\alpha x^2} \, dx = \sqrt{\frac{\pi}{\alpha}}.
     \]
   - For \( n = 2 \) (i.e., \( k = 1 \)): 
     \[
     I_2 = \int_{-\infty}^{\infty} x^2 e^{-\alpha x^2} \, dx = \frac{1}{2\alpha} \sqrt{\frac{\pi}{\alpha}}.
     \]
   - For \( n = 4 \) (i.e., \( k = 2 \)): 
     \[
     I_4 = \frac{3}{(2\alpha)^2} \sqrt{\frac{\pi}{\alpha}} = \frac{3}{4\alpha^2} \sqrt{\frac{\pi}{\alpha}}.
     \]

Thus, 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 more details or have any questions about this? Here are some related questions:

1. What is the derivation process for this Gaussian integral?
2. How do odd and even functions affect integrals over symmetric intervals?
3. What is the significance of the double factorial in combinatorics and integrals?
4. How can Gaussian integrals be used in probability theory and physics?
5. How does the result change if the limits of integration are not symmetric?

**Tip:** Recognizing even and odd functions can simplify many integrals, especially those involving symmetric intervals.

Determine the general formula for the integral \( \int_{-\infty}^{\infty} x^n e^{-\alpha x^2} \, dx \) for n = 0, 1, 2, 3, 4, \dots.

Find the general formula for the Gaussian integral \( \int_{-\infty}^{\infty} x^n e^{-\alpha x^2} \, dx \) for both even and odd powers of n. The solution includes key mathematical concepts like Gaussian integrals, even and odd function properties, and double factorials.

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