1. **State the problem:** We need to find the indefinite integral $$\int (x^2 - 5)^3 \, dx$$. 2. **Formula and approach:** To integrate a composite function raised to a power, we can use substitution. Let $$u = x^2 - 5$$, then $$du = 2x \, dx$$. 3. **Important note:** Our integral is in terms of $$x$$, but substitution requires $$du$$ to be present. Since $$du = 2x \, dx$$, and our integral does not have an $$x$$ term outside, we will expand the integrand instead. 4. **Expand the integrand:** $$(x^2 - 5)^3 = (x^2 - 5)(x^2 - 5)(x^2 - 5)$$ First, expand two factors: $$(x^2 - 5)(x^2 - 5) = x^4 - 10x^2 + 25$$ Now multiply by the third factor: $$(x^4 - 10x^2 + 25)(x^2 - 5) = x^6 - 5x^4 - 10x^4 + 50x^2 + 25x^2 - 125$$ Simplify terms: $$x^6 - 15x^4 + 75x^2 - 125$$ 5. **Rewrite the integral:** $$\int (x^2 - 5)^3 \, dx = \int (x^6 - 15x^4 + 75x^2 - 125) \, dx$$ 6. **Integrate term-by-term:** $$\int x^6 \, dx = \frac{x^7}{7}$$ $$\int (-15x^4) \, dx = -15 \cdot \frac{x^5}{5} = -3x^5$$ $$\int 75x^2 \, dx = 75 \cdot \frac{x^3}{3} = 25x^3$$ $$\int (-125) \, dx = -125x$$ 7. **Combine results:** $$\int (x^2 - 5)^3 \, dx = \frac{x^7}{7} - 3x^5 + 25x^3 - 125x + C$$ **Final answer:** $$\boxed{\frac{x^7}{7} - 3x^5 + 25x^3 - 125x + C}$$