1. **State the problem:** We need to find the indefinite integral $$\int 18x^2 (x^3+2)^5 \, dx.$$\n\n2. **Identify the method:** This integral suggests using substitution because the integrand contains a composite function $(x^3+2)^5$ and its derivative $3x^2$ appears as a factor.\n\n3. **Choose substitution:** Let $$u = x^3 + 2.$$ Then, differentiate both sides with respect to $x$:\n$$\frac{du}{dx} = 3x^2 \implies du = 3x^2 \, dx.$$\n\n4. **Rewrite the integral:** We have $18x^2 \, dx$ in the integral, so express it in terms of $du$:\n$$18x^2 \, dx = 6 \cdot 3x^2 \, dx = 6 \, du.$$\n\n5. **Substitute into the integral:**\n$$\int 18x^2 (x^3+2)^5 \, dx = \int 6 u^5 \, du.$$\n\n6. **Integrate with respect to $u$:**\n$$\int 6 u^5 \, du = 6 \int u^5 \, du = 6 \cdot \frac{u^6}{6} = u^6 + C,$$\nwhere $C$ is the constant of integration.\n\n7. **Back-substitute $u$:**\n$$u^6 + C = (x^3 + 2)^6 + C.$$\n\n**Final answer:**\n$$\int 18x^2 (x^3+2)^5 \, dx = (x^3 + 2)^6 + C.$$