1. **Problem Statement:** Evaluate the integral $$\int (2x^3 + 1)^7 x^2 \, dx$$ using the change of variable method. 2. **Recall the formula:** For integrals of the form $$\int (g(x))^n g'(x) \, dx = \frac{(g(x))^{n+1}}{n+1} + C$$ where $g(x)$ is a differentiable function and $n \neq -1$. 3. **Identify $g(x)$ and $g'(x)$:** Let $$g(x) = 2x^3 + 1$$ then $$g'(x) = 6x^2$$. 4. **Rewrite the integral:** The integral is $$\int (2x^3 + 1)^7 x^2 \, dx = \int (g(x))^7 x^2 \, dx$$. 5. **Express $x^2 dx$ in terms of $dg$:** Since $$dg = g'(x) dx = 6x^2 dx$$, then $$x^2 dx = \frac{dg}{6}$$. 6. **Substitute into the integral:** $$\int (g(x))^7 x^2 \, dx = \int g^7 \cdot \frac{dg}{6} = \frac{1}{6} \int g^7 \, dg$$. 7. **Integrate with respect to $g$:** $$\frac{1}{6} \int g^7 \, dg = \frac{1}{6} \cdot \frac{g^{8}}{8} + C = \frac{g^{8}}{48} + C$$. 8. **Back-substitute $g(x)$:** $$\frac{(2x^3 + 1)^8}{48} + C$$. **Final answer:** $$\int (2x^3 + 1)^7 x^2 \, dx = \frac{(2x^3 + 1)^8}{48} + C$$