The problem involves solving the integral $\int 3x e^{x^2} \, dx$. To solve this, we use the technique of substitution rather than moving constants outside, as suggested in the text.

Step-by-step Solution:

1. Substitution: Let $u = x^2$. Then, the derivative is: $du = 2x \, dx \quad \text{or} \quad x \, dx = \frac{1}{2} \, du.$

2. Rewrite the Integral: Substituting $u = x^2$ and $x \, dx = \frac{1}{2} \, du$, the integral becomes: $\int 3x e^{x^2} \, dx = 3 \int e^u \cdot \frac{1}{2} \, du = \frac{3}{2} \int e^u \, du.$

3. Integrate: The integral of $e^u$ is simply $e^u$. So: $\frac{3}{2} \int e^u \, du = \frac{3}{2} e^u + C.$

4. Back-Substitute $u = x^2$: Replace $u$ with $x^2$ to get the final answer: $\frac{3}{2} e^{x^2} + C.$

Final Answer:

$\boxed{\frac{3}{2} e^{x^2} + C}$

Let me know if you want further explanation or details!

Related Questions:

1. What is the general method for solving integrals of the form $\int x e^{f(x)} \, dx$?
2. How does substitution simplify solving integrals?
3. Can integration by parts also solve this problem? If yes, how?
4. Why does moving the constant outside not help in this specific case?
5. What are the advantages of substitution compared to other methods?

Tip:

Always look for patterns in integrals involving $e^{x^2}$ or similar expressions. Substitution is often the fastest method if the derivative of the inner function is present in the integrand.