1. **Problem Statement:** Evaluate the integral $$\int x e^{x^2} \, dx$$. 2. **Choosing the Technique:** This integral suggests using substitution because the exponent $x^2$ is a function whose derivative appears as a factor ($x$) outside the exponential. 3. **Substitution:** Let $$u = x^2$$. Then, $$du = 2x \, dx \implies \frac{du}{2} = x \, dx$$. 4. **Rewrite the Integral:** Substitute into the integral: $$\int x e^{x^2} \, dx = \int e^u \frac{du}{2} = \frac{1}{2} \int e^u \, du$$. 5. **Integrate:** The integral of $e^u$ with respect to $u$ is $e^u$: $$\frac{1}{2} \int e^u \, du = \frac{1}{2} e^u + C$$. 6. **Back-substitute:** Replace $u$ with $x^2$: $$\frac{1}{2} e^{x^2} + C$$. 7. **Summary:** We used substitution because the derivative of the inner function $x^2$ appeared in the integrand. This technique simplifies the integral to a basic exponential integral. **Final answer:** $$\boxed{\frac{1}{2} e^{x^2} + C}$$