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 method:** To integrate a rational function with a quadratic denominator, complete the square in the denominator. 3. **Complete the square:** $$x^2 + x + 1 = \left(x + \frac{1}{2}\right)^2 + 1 - \frac{1}{4} = \left(x + \frac{1}{2}\right)^2 + \frac{3}{4}$$ 4. **Rewrite the integral:** $$\int_1^{10} \frac{2624}{\left(x + \frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} \, dx$$ 5. **Use substitution:** Let $$u = x + \frac{1}{2}$$, so $$du = dx$$. 6. **Change limits:** When $$x=1$$, $$u=1 + \frac{1}{2} = \frac{3}{2}$$; when $$x=10$$, $$u=10 + \frac{1}{2} = \frac{21}{2}$$. 7. **Integral becomes:** $$\int_{\frac{3}{2}}^{\frac{21}{2}} \frac{2624}{u^2 + \left(\frac{\sqrt{3}}{2}\right)^2} \, du$$ 8. **Recall the integral formula:** $$\int \frac{1}{u^2 + a^2} du = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C$$ 9. **Apply the formula:** Here, $$a = \frac{\sqrt{3}}{2}$$, so $$\int_{\frac{3}{2}}^{\frac{21}{2}} \frac{2624}{u^2 + a^2} du = 2624 \times \frac{1}{a} \left[ \arctan\left(\frac{u}{a}\right) \right]_{\frac{3}{2}}^{\frac{21}{2}}$$ 10. **Calculate:** $$= 2624 \times \frac{1}{\frac{\sqrt{3}}{2}} \left( \arctan\left( \frac{\frac{21}{2}}{\frac{\sqrt{3}}{2}} \right) - \arctan\left( \frac{\frac{3}{2}}{\frac{\sqrt{3}}{2}} \right) \right)$$ 11. **Simplify the fractions inside arctan:** $$\frac{\frac{21}{2}}{\frac{\sqrt{3}}{2}} = \frac{21}{\sqrt{3}} = 7 \sqrt{3}$$ $$\frac{\frac{3}{2}}{\frac{\sqrt{3}}{2}} = \frac{3}{\sqrt{3}} = \sqrt{3}$$ 12. **Simplify the coefficient:** $$\frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}$$ 13. **Final expression:** $$= 2624 \times \frac{2}{\sqrt{3}} \left( \arctan(7 \sqrt{3}) - \arctan(\sqrt{3}) \right)$$ 14. **Evaluate arctan values:** - $$\arctan(\sqrt{3}) = \frac{\pi}{3}$$ - $$\arctan(7 \sqrt{3})$$ is an angle but we leave it as is for exact form. 15. **Therefore, the exact value is:** $$\boxed{\frac{5248}{\sqrt{3}} \left( \arctan(7 \sqrt{3}) - \frac{\pi}{3} \right)}$$ This is the exact value of the definite integral.