1. The problem is to evaluate the integral $$\int (x^2 - 5)^3 \, dx$$. 2. We use the substitution method for integration. Let $$u = x^2 - 5$$. Then, the derivative is $$\frac{du}{dx} = 2x$$, or $$du = 2x \, dx$$. 3. Notice that the integral does not have an $$x$$ term outside the power, so direct substitution is not straightforward. Instead, we expand the integrand before integrating. 4. Expand $$(x^2 - 5)^3$$ using the binomial theorem: $$ (x^2 - 5)^3 = (x^2)^3 - 3 \cdot (x^2)^2 \cdot 5 + 3 \cdot x^2 \cdot 5^2 - 5^3 = x^6 - 15x^4 + 75x^2 - 125 $$ 5. Now, 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. Compute 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. Substitute back: $$ \frac{x^7}{7} - 15 \cdot \frac{x^5}{5} + 75 \cdot \frac{x^3}{3} - 125x + C = \frac{x^7}{7} - 3x^5 + 25x^3 - 125x + C $$ 8. The options given do not match this expanded integral result, so let's try substitution again with a different approach. 9. Using substitution $$u = x^2 - 5$$, then $$du = 2x \, dx$$, so $$dx = \frac{du}{2x}$$. 10. The integral becomes: $$ \int u^3 \, dx = \int u^3 \cdot \frac{du}{2x} $$, which is not directly integrable because of the $$x$$ in the denominator. 11. Since the options suggest an answer in terms of $$(x^2 - 5)^n$$, let's try differentiating the options to see which derivative matches the integrand. 12. Differentiate option d: $$F(x) = \frac{(x^2 - 5)^4}{8} + C$$ $$F'(x) = \frac{4(x^2 - 5)^3 \cdot 2x}{8} = (x^2 - 5)^3 x$$ 13. The derivative has an extra $$x$$ factor, but our integrand is $$(x^2 - 5)^3$$ without $$x$$. 14. This suggests the integral cannot be expressed simply as a function of $$(x^2 - 5)^n$$ without an $$x$$ factor. 15. Therefore, the correct integral is the expanded form: $$ \int (x^2 - 5)^3 \, dx = \frac{x^7}{7} - 3x^5 + 25x^3 - 125x + C $$ 16. None of the options a, b, c, or d match this result exactly. Final answer: $$\int (x^2 - 5)^3 \, dx = \frac{x^7}{7} - 3x^5 + 25x^3 - 125x + C$$