To solve the integral $\int x \cdot e^{-x^2} \, dx$, we can use substitution.

Step-by-step solution:

1. Substitute $u = x^2$, so that $du = 2x \, dx$, or equivalently, $\frac{du}{2} = x \, dx$.

2. This changes the integral to: $\int x \cdot e^{-x^2} \, dx = \int e^{-u} \cdot \frac{du}{2}$

3. The integral now becomes: $\frac{1}{2} \int e^{-u} \, du$

4. The integral of $e^{-u}$ is $-e^{-u}$, so we get: $\frac{1}{2} (-e^{-u}) = -\frac{1}{2} e^{-u}$

5. Finally, substitute back $u = x^2$ to get the solution in terms of $x$: $-\frac{1}{2} e^{-x^2} + C$

Thus, the integral is: $\int x \cdot e^{-x^2} \, dx = -\frac{1}{2} e^{-x^2} + C$

---

Would you like a deeper explanation of any step, or do you have another question? Here's some follow-up content:

1. How would the substitution method change if the integral were $\int x^3 \cdot e^{-x^2} \, dx$?
2. What would happen if the exponent in $e^{-x^2}$ were changed to a different quadratic form, like $e^{-2x^2}$?
3. How does integration by parts compare to substitution in solving integrals like this?
4. What are other examples of integrals involving exponential functions that can be solved using substitution?
5. Could this integral be related to a Gaussian integral or error function in some way?

Tip: When using substitution, always check if the derivative of one part of the integrand matches another part. This is key to making a good substitution.