1. **State the problem:** Evaluate the indefinite integral $$\int x \sqrt{8 - x^2} \, dx$$. 2. **Recall the formula and substitution method:** When integrating expressions involving $x$ and $\sqrt{a - x^2}$, a common approach is to use substitution. Here, let $$u = 8 - x^2$$. 3. **Compute the differential:** Differentiating $u$ with respect to $x$ gives $$\frac{du}{dx} = -2x \implies du = -2x \, dx \implies x \, dx = -\frac{1}{2} du$$. 4. **Rewrite the integral in terms of $u$:** Substitute $x \, dx$ and $\sqrt{8 - x^2}$: $$\int x \sqrt{8 - x^2} \, dx = \int \sqrt{u} \left(-\frac{1}{2} du\right) = -\frac{1}{2} \int u^{1/2} \, du$$. 5. **Integrate with respect to $u$:** Use the power rule for integration: $$\int u^{1/2} \, du = \frac{u^{3/2}}{\frac{3}{2}} = \frac{2}{3} u^{3/2}$$. 6. **Substitute back to $x$:** $$-\frac{1}{2} \times \frac{2}{3} u^{3/2} + C = -\frac{1}{3} (8 - x^2)^{3/2} + C$$. 7. **Final answer:** $$\int x \sqrt{8 - x^2} \, dx = -\frac{1}{3} (8 - x^2)^{3/2} + C$$. This completes the evaluation of the integral using substitution and the power rule.