1. **State the problem:** Evaluate the integral $$\int (x^2 - 5)^3 \, dx$$. 2. **Formula and substitution:** Use substitution for integrals of composite functions. Let $$u = x^2 - 5$$, then $$\frac{du}{dx} = 2x$$ or $$du = 2x \, dx$$. 3. **Rewrite the integral:** The integral becomes $$\int u^3 \, dx$$, but we need to express $$dx$$ in terms of $$du$$ and $$x$$. Since $$du = 2x \, dx$$, we have $$dx = \frac{du}{2x}$$. 4. **Problem with substitution:** The integral $$\int (x^2 - 5)^3 \, dx$$ does not directly fit the substitution because of the $$x$$ outside the substitution. Instead, expand the integrand: $$(x^2 - 5)^3 = (x^2)^3 - 3 \cdot (x^2)^2 \cdot 5 + 3 \cdot x^2 \cdot 5^2 - 5^3 = x^6 - 15x^4 + 75x^2 - 125$$ 5. **Integrate term-by-term:** $$\int (x^6 - 15x^4 + 75x^2 - 125) \, dx = \int x^6 \, dx - 15 \int x^4 \, dx + 75 \int x^2 \, dx - 125 \int dx$$ 6. **Calculate each integral:** $$\int x^6 \, dx = \frac{x^7}{7}$$ $$\int x^4 \, dx = \frac{x^5}{5}$$ $$\int x^2 \, dx = \frac{x^3}{3}$$ $$\int dx = x$$ 7. **Combine results:** $$\frac{x^7}{7} - 15 \cdot \frac{x^5}{5} + 75 \cdot \frac{x^3}{3} - 125x + C = \frac{x^7}{7} - 3x^5 + 25x^3 - 125x + C$$ 8. **Check answer choices:** None of the options match this expanded integral, so the problem likely expects a substitution approach with a different method. 9. **Alternative approach:** Use substitution with $$u = x^2 - 5$$, then $$du = 2x \, dx$$. Rewrite the integral as: $$\int (x^2 - 5)^3 \, dx = \int u^3 \, dx$$ But since $$du = 2x \, dx$$, we need an $$x$$ in the integrand to substitute properly. The integral lacks an $$x$$ factor, so direct substitution is not straightforward. 10. **Use integration by parts or expand:** Since substitution is complicated, the best approach is expansion as done. **Final answer:** $$\int (x^2 - 5)^3 \, dx = \frac{x^7}{7} - 3x^5 + 25x^3 - 125x + C$$. **None of the given options a, b, c, d match this result.** --- **Summary:** The integral $$\int (x^2 - 5)^3 \, dx$$ is best solved by expanding the integrand and integrating term-by-term, resulting in $$\frac{x^7}{7} - 3x^5 + 25x^3 - 125x + C$$.