https://file.aiquickdraw.com/mathbot/file/172971754042596idlzo9.jpg

The image contains three integrals that need to be evaluated. Let's break down each of them and solve them one by one:

1. **Integral 1:**

\[
\int_0^{+\infty} x e^{-x^2} \, dx
\]

This integral is a standard Gaussian integral that can be solved using a substitution method.

- **Substitution**: Let \( u = x^2 \), then \( du = 2x \, dx \), or \( x \, dx = \frac{1}{2} \, du \).
  
The limits change accordingly:
- When \( x = 0 \), \( u = 0 \),
- When \( x = +\infty \), \( u = +\infty \).

Thus, the integral becomes:

\[
\int_0^{+\infty} x e^{-x^2} \, dx = \frac{1}{2} \int_0^{+\infty} e^{-u} \, du
\]

\[
= \frac{1}{2} \left[ -e^{-u} \right]_0^{+\infty} = \frac{1}{2} \left( 0 - (-1) \right) = \frac{1}{2}
\]

So, the result of the first integral is \( \frac{1}{2} \).

---

2. **Integral 2:**

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

This is a symmetric integral because the function \( x e^{-x^2} \) is **odd** with respect to \( x \). The integral of an odd function over symmetric limits (from \(-\infty\) to \(+\infty\)) is always zero. Thus, we have:

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

---

3. **Integral 3:**

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

This integral is problematic because the exponential function \( e^{x^2} \) grows very rapidly as \( x \to \pm \infty \), and this makes the integral diverge. In other words, this integral does not converge to a finite value. Hence, the result is:

\[
\text{Divergent}
\]

---

### Summary of Results:
1. \( \int_0^{+\infty} x e^{-x^2} \, dx = \frac{1}{2} \)
2. \( \int_{-\infty}^{+\infty} x e^{-x^2} \, dx = 0 \)
3. \( \int_{-\infty}^{+\infty} x e^{x^2} \, dx \) is divergent.

Would you like more details on any of these steps? Feel free to ask!

### Related Questions:
1. How can you solve integrals involving other Gaussian-like functions?
2. What are the properties of odd and even functions in integrals?
3. How do you handle divergent integrals in practice?
4. What other techniques can be used for solving Gaussian integrals?
5. How do substitution techniques help in simplifying integrals?

### Tip:
For integrals involving Gaussian functions, substitution techniques and recognizing patterns like odd/even functions are crucial for simplifying the work.

Evaluate the following integrals: 1. ∫₀⁺∞ x e^(-x²) dx, 2. ∫₋∞⁺∞ x e^(-x²) dx, 3. ∫₋∞⁺∞ x e^(x²) dx.

This problem involves solving Gaussian integrals, exploring odd functions, and understanding why certain integrals diverge. Key concepts include substitution methods and properties of symmetric integrals, suitable for university-level calculus students.

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