1. **State the problem:** We need to evaluate the definite integral $$\int_1^{10} \frac{2624}{x^2 + x + 1} \, dx$$. 2. **Recall the formula and approach:** The integral of the form $$\int \frac{1}{x^2 + bx + c} \, dx$$ can be solved by completing the square in the denominator and using the arctangent formula: $$\int \frac{dx}{(x + \frac{b}{2})^2 + (c - \frac{b^2}{4})} = \frac{1}{\sqrt{c - \frac{b^2}{4}}} \arctan \left( \frac{x + \frac{b}{2}}{\sqrt{c - \frac{b^2}{4}}} \right) + C$$ 3. **Complete the square for the denominator:** $$x^2 + x + 1 = \left(x + \frac{1}{2}\right)^2 + 1 - \frac{1}{4} = \left(x + \frac{1}{2}\right)^2 + \frac{3}{4}$$ 4. **Rewrite the integral:** $$\int_1^{10} \frac{2624}{\left(x + \frac{1}{2}\right)^2 + \frac{3}{4}} \, dx = 2624 \int_1^{10} \frac{dx}{\left(x + \frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2}$$ 5. **Use substitution:** Let $$u = x + \frac{1}{2}$$, so when $$x=1$$, $$u=1.5$$, and when $$x=10$$, $$u=10.5$$. 6. **Apply the arctangent integral formula:** $$\int \frac{du}{u^2 + a^2} = \frac{1}{a} \arctan \left( \frac{u}{a} \right) + C$$ where $$a = \frac{\sqrt{3}}{2}$$. 7. **Evaluate the definite integral:** $$2624 \times \frac{1}{\frac{\sqrt{3}}{2}} \left[ \arctan \left( \frac{u}{\frac{\sqrt{3}}{2}} \right) \right]_{1.5}^{10.5} = 2624 \times \frac{2}{\sqrt{3}} \left[ \arctan \left( \frac{10.5}{\frac{\sqrt{3}}{2}} \right) - \arctan \left( \frac{1.5}{\frac{\sqrt{3}}{2}} \right) \right]$$ 8. **Simplify the arguments:** $$\frac{10.5}{\frac{\sqrt{3}}{2}} = \frac{10.5 \times 2}{\sqrt{3}} = \frac{21}{\sqrt{3}}$$ $$\frac{1.5}{\frac{\sqrt{3}}{2}} = \frac{1.5 \times 2}{\sqrt{3}} = \frac{3}{\sqrt{3}} = \sqrt{3}$$ 9. **Final expression:** $$\int_1^{10} \frac{2624}{x^2 + x + 1} \, dx = \frac{5248}{\sqrt{3}} \left( \arctan \left( \frac{21}{\sqrt{3}} \right) - \arctan (\sqrt{3}) \right)$$ 10. **Numerical approximation:** $$\arctan(\sqrt{3}) = \frac{\pi}{3} \approx 1.0472$$ $$\arctan \left( \frac{21}{\sqrt{3}} \right) \approx 1.523$$ So, $$\int_1^{10} \frac{2624}{x^2 + x + 1} \, dx \approx \frac{5248}{1.732} (1.523 - 1.0472) = 3028.6 \times 0.4758 \approx 1440.5$$ **Answer:** $$\boxed{1440.5}$$