1. **State the problem:** Evaluate the definite integral $$\int_1^{10} \frac{2624}{x^2 - x - 1} \, dx$$. 2. **Recall the formula and approach:** To integrate a rational function where the denominator is a quadratic, we first factor or complete the square if possible, or use partial fraction decomposition if the quadratic factors. 3. **Factor the denominator:** The quadratic is $$x^2 - x - 1$$. The roots are given by the quadratic formula: $$x = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}$$. So, $$x^2 - x - 1 = (x - \frac{1 + \sqrt{5}}{2})(x - \frac{1 - \sqrt{5}}{2})$$. 4. **Set up partial fractions:** $$\frac{2624}{(x - \frac{1 + \sqrt{5}}{2})(x - \frac{1 - \sqrt{5}}{2})} = \frac{A}{x - \frac{1 + \sqrt{5}}{2}} + \frac{B}{x - \frac{1 - \sqrt{5}}{2}}$$ Multiply both sides by the denominator: $$2624 = A\left(x - \frac{1 - \sqrt{5}}{2}\right) + B\left(x - \frac{1 + \sqrt{5}}{2}\right)$$. 5. **Find A and B:** Set $$x = \frac{1 + \sqrt{5}}{2}$$: $$2624 = A\left(\frac{1 + \sqrt{5}}{2} - \frac{1 - \sqrt{5}}{2}\right) + B \cdot 0 = A(\sqrt{5})$$ $$\Rightarrow A = \frac{2624}{\sqrt{5}}$$. Set $$x = \frac{1 - \sqrt{5}}{2}$$: $$2624 = A \cdot 0 + B\left(\frac{1 - \sqrt{5}}{2} - \frac{1 + \sqrt{5}}{2}\right) = B(-\sqrt{5})$$ $$\Rightarrow B = -\frac{2624}{\sqrt{5}}$$. 6. **Rewrite the integral:** $$\int_1^{10} \frac{2624}{x^2 - x - 1} \, dx = \int_1^{10} \frac{2624/\sqrt{5}}{x - \frac{1 + \sqrt{5}}{2}} - \frac{2624/\sqrt{5}}{x - \frac{1 - \sqrt{5}}{2}} \, dx$$. 7. **Integrate term-by-term:** $$= \frac{2624}{\sqrt{5}} \int_1^{10} \frac{1}{x - \frac{1 + \sqrt{5}}{2}} \, dx - \frac{2624}{\sqrt{5}} \int_1^{10} \frac{1}{x - \frac{1 - \sqrt{5}}{2}} \, dx$$ 8. **Integrals of the form $$\int \frac{1}{x - a} dx = \ln|x - a| + C$$:** So, $$= \frac{2624}{\sqrt{5}} \left[ \ln\left|x - \frac{1 + \sqrt{5}}{2}\right| \right]_1^{10} - \frac{2624}{\sqrt{5}} \left[ \ln\left|x - \frac{1 - \sqrt{5}}{2}\right| \right]_1^{10}$$ 9. **Evaluate the definite integrals:** $$= \frac{2624}{\sqrt{5}} \left( \ln\left|10 - \frac{1 + \sqrt{5}}{2}\right| - \ln\left|1 - \frac{1 + \sqrt{5}}{2}\right| \right) - \frac{2624}{\sqrt{5}} \left( \ln\left|10 - \frac{1 - \sqrt{5}}{2}\right| - \ln\left|1 - \frac{1 - \sqrt{5}}{2}\right| \right)$$ 10. **Simplify using log properties:** $$= \frac{2624}{\sqrt{5}} \left( \ln\frac{10 - \frac{1 + \sqrt{5}}{2}}{1 - \frac{1 + \sqrt{5}}{2}} - \ln\frac{10 - \frac{1 - \sqrt{5}}{2}}{1 - \frac{1 - \sqrt{5}}{2}} \right)$$ 11. **Calculate the numerical values inside the logs:** - $$10 - \frac{1 + \sqrt{5}}{2} = 10 - 1.618... = 8.382...$$ - $$1 - \frac{1 + \sqrt{5}}{2} = 1 - 1.618... = -0.618...$$ - $$10 - \frac{1 - \sqrt{5}}{2} = 10 - (-0.618...) = 10.618...$$ - $$1 - \frac{1 - \sqrt{5}}{2} = 1 - (-0.618...) = 1.618...$$ 12. **Plug in and simplify:** $$= \frac{2624}{\sqrt{5}} \left( \ln\left|\frac{8.382}{-0.618}\right| - \ln\left|\frac{10.618}{1.618}\right| \right) = \frac{2624}{\sqrt{5}} \left( \ln(13.56) - \ln(6.56) \right)$$ 13. **Final simplification:** $$= \frac{2624}{\sqrt{5}} \ln\left( \frac{13.56}{6.56} \right) = \frac{2624}{\sqrt{5}} \ln(2.067)$$ 14. **Calculate the approximate value:** $$\sqrt{5} \approx 2.236$$ $$\ln(2.067) \approx 0.726$$ So, $$\approx \frac{2624}{2.236} \times 0.726 = 1173.3 \,.$$