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.