$$ \int_{}^{}\!x^{2r}\cdot\,e^{-\frac{x^2}{2\sigma^2}}\,\mathrm{d}x $$ where r and sigma are constants

The given integral is:

\[
I = \int x^{2r} e^{-\frac{x^2}{2\sigma^2}} \, dx
\]

where $ r $ and $ \sigma $ are constants. This resembles a Gaussian integral with an additional factor $ x^{2r} $. 

To solve this, we use the general result for integrals of the form:

\[
\int_{-\infty}^{\infty} x^{2r} e^{-\frac{x^2}{2\sigma^2}} \, dx = (2\sigma^2)^r \cdot \sqrt{2\pi \sigma^2} \cdot \frac{(2r-1)!!}{2^r}
\]

where $ (2r-1)!! $ is the double factorial, which is defined as:

\[
(2r-1)!! = (2r-1)(2r-3)(2r-5)\dots(1)
\]

### Full Steps:

1. **Substitute**: Start by substituting $ u = \frac{x^2}{2\sigma^2} $, then simplify the exponential term.
   
2. **General Gaussian Integral Formula**: Apply the known formula for integrals of Gaussian functions with powers of $ x $.

3. **Solution**:
   \[
   I = (2\sigma^2)^r \cdot \sqrt{2\pi \sigma^2} \cdot \frac{(2r-1)!!}{2^r}
   \]
   This provides the result in terms of $ r $ and $ \sigma $.

Would you like further details or clarifications on any part?

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Here are 5 related questions:
1. How is the double factorial defined for odd and even numbers?
2. Can the same technique be applied if the exponent on $x$ is an odd number instead of $2r$?
3. How does the presence of the Gaussian exponential influence the integrability of the function?
4. What happens to the integral when $ r = 0 $?
5. How does changing the constant $ \sigma $ affect the result of the integral?

**Tip**: Gaussian integrals are highly symmetric and appear frequently in probability theory, especially in the context of normal distributions.

$$ \int_{}^{} x^{2r} \cdot e^{-\frac{x^2}{2\sigma^2}} \, dx $$ where r and sigma are constants

Learn how to solve the Gaussian integral of the form $$ \int x^{2r} e^{-\frac{x^2}{2\sigma^2}} \, dx $$ where r and sigma are constants. This solution uses advanced mathematical techniques like Gaussian integrals and double factorials, commonly applied in probability theory and mathematical physics.

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