1. **State the problem:** We are asked to evaluate the integral $$\int (3x - 24) \, dx$$ and also consider the substitution $$x = 2y$$. 2. **Recall the integral formula:** The integral of a polynomial function $$\int (ax + b) \, dx = \frac{a}{2}x^2 + bx + C$$ where $$C$$ is the constant of integration. 3. **Evaluate the integral directly:** $$\int (3x - 24) \, dx = \int 3x \, dx - \int 24 \, dx = \frac{3}{2}x^2 - 24x + C$$ 4. **Apply the substitution $$x = 2y$$:** Substitute $$x = 2y$$ into the integral expression: $$\int (3(2y) - 24) \, d(2y)$$ Since $$dx = 2 dy$$, rewrite the integral in terms of $$y$$: $$\int (6y - 24) \cdot 2 \, dy = \int (12y - 48) \, dy$$ 5. **Evaluate the integral in terms of $$y$$:** $$\int (12y - 48) \, dy = 6y^2 - 48y + C$$ 6. **Rewrite the result back in terms of $$x$$:** Since $$y = \frac{x}{2}$$, $$6\left(\frac{x}{2}\right)^2 - 48 \left(\frac{x}{2}\right) + C = 6 \cdot \frac{x^2}{4} - 24x + C = \frac{3}{2}x^2 - 24x + C$$ 7. **Conclusion:** Both direct integration and substitution yield the same result: $$\boxed{\int (3x - 24) \, dx = \frac{3}{2}x^2 - 24x + C}$$ This confirms the integral evaluation is correct.