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The integral you provided is:

\[
\int x e^{-x^2} \, dx
\]

with the substitution \( u = -x^2 \).

Let's solve this step by step using the substitution method:

### Step 1: Substitution
Given:
\[
u = -x^2
\]
Differentiate \( u \) with respect to \( x \) to find \( du \):
\[
du = -2x \, dx
\]

This implies:
\[
\frac{du}{-2} = x \, dx
\]

### Step 2: Substitute into the Integral
Now substitute into the integral:
\[
\int x e^{-x^2} \, dx = \int e^u \cdot \frac{du}{-2}
\]

This simplifies to:
\[
-\frac{1}{2} \int e^u \, du
\]

### Step 3: Integrate
The integral of \( e^u \) with respect to \( u \) is simply \( e^u \):
\[
-\frac{1}{2} \int e^u \, du = -\frac{1}{2} e^u + C
\]

### Step 4: Substitute Back
Finally, substitute back \( u = -x^2 \):
\[
-\frac{1}{2} e^u + C = -\frac{1}{2} e^{-x^2} + C
\]

Thus, the solution to the integral is:
\[
\int x e^{-x^2} \, dx = -\frac{1}{2} e^{-x^2} + C
\]

Would you like more details on any part of the solution, or do you have any questions?

Here are some related questions you might find interesting:
1. How does the substitution method work in integrals?
2. What are the common substitutions used for Gaussian integrals?
3. Can you explain how integration by parts might apply to a similar problem?
4. What are other techniques to solve integrals involving exponentials and polynomials?
5. How do definite integrals differ when using substitution?

**Tip:** Always check if substitution simplifies the integral by directly transforming the differential \( dx \) and integrand.

Evaluate the integral: ∫ xe^(-x^2) dx, using u = -x^2

Learn how to solve the integral ∫ xe^(-x^2) dx by using the substitution method (u = -x^2). This solution provides a step-by-step explanation of the integral process and is suitable for students studying undergraduate calculus.

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