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Determine and outline the steps to determine whether the integral is convergent or divergent.

The given integral is:

\[
\int_{-\infty}^{\infty} x^3 e^{-x^2} \, dx
\]

To determine whether this integral is convergent or divergent, let's go through the following steps:

### 1. **Check the integrand's behavior at infinity:**
   - The function inside the integral is \(x^3 e^{-x^2}\).
   - As \( x \to \infty \) or \( x \to -\infty \), the term \( e^{-x^2} \) decays very rapidly to zero because it is an exponentially decaying function.
   - However, \( x^3 \) grows for large values of \( |x| \), but the decay of \( e^{-x^2} \) is much faster than the growth of \( x^3 \).
   - Therefore, the integrand tends to zero at both \( +\infty \) and \( -\infty \).

### 2. **Symmetry of the integrand:**
   - The function \( x^3 \) is **odd**, meaning \( x^3 = -(-x)^3 \).
   - The exponential function \( e^{-x^2} \) is **even**, meaning \( e^{-x^2} = e^{-(-x)^2} \).
   - Therefore, the integrand \( x^3 e^{-x^2} \) is **odd** because it is the product of an odd and an even function.
   - Integrals of odd functions over symmetric limits (from \( -\infty \) to \( \infty \)) are zero.

### 3. **Conclusion:**
   - Since the integrand is odd and we are integrating over a symmetric interval (from \( -\infty \) to \( \infty \)), the integral converges and evaluates to **0**.

Thus, the integral is **convergent** and its value is:

\[
\int_{-\infty}^{\infty} x^3 e^{-x^2} \, dx = 0
\]

Let me know if you would like further details!

Here are five related questions for further practice:
1. What is the result of \(\int_{-\infty}^{\infty} x^2 e^{-x^2} \, dx\)?
2. How do you evaluate \(\int_{0}^{\infty} e^{-x} \, dx\)?
3. How would the integral change if the power of \(x\) were even, e.g., \(x^2\)?
4. What is the value of \(\int_{-\infty}^{\infty} e^{-x^2} \, dx\)?
5. How do you handle improper integrals where the limits are infinite?

**Tip:** Integrals involving Gaussian functions like \( e^{-x^2} \) often have closed-form solutions and are useful in probability and statistics!

Learn how to determine whether the improper integral ∫ x^3 * e^(-x^2) dx from -∞ to ∞ is convergent or divergent. This solution explores the behavior of the integrand at infinity, the symmetry of the function, and provides a detailed explanation of why the integral converges to 0.

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