1. **State the problem:** Evaluate the indefinite integral $$\int x^2 \sqrt{x^3 + 7} \, dx$$ using the substitution $$u = x^3 + 7$$. 2. **Identify the substitution and its derivative:** Given $$u = x^3 + 7$$, differentiate both sides with respect to $$x$$: $$\frac{du}{dx} = 3x^2 \implies du = 3x^2 \, dx$$. 3. **Rewrite the integral in terms of $$u$$:** Solve for $$x^2 \, dx$$: $$x^2 \, dx = \frac{du}{3}$$. 4. **Substitute into the integral:** Replace $$x^3 + 7$$ with $$u$$ and $$x^2 \, dx$$ with $$\frac{du}{3}$$: $$\int x^2 \sqrt{x^3 + 7} \, dx = \int \sqrt{u} \cdot \frac{du}{3} = \frac{1}{3} \int u^{1/2} \, du$$. 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. **Combine constants and write the result:** $$\frac{1}{3} \times \frac{2}{3} u^{3/2} + C = \frac{2}{9} u^{3/2} + C$$. 7. **Back-substitute $$u$$:** Replace $$u$$ with $$x^3 + 7$$: $$\boxed{\frac{2}{9} (x^3 + 7)^{3/2} + C}$$. This is the evaluated indefinite integral using the substitution method.