1. **State the problem:** Calculate the definite integral $$\int_1^{10} \frac{2621}{x^2 - x - 1} \, dx$$. 2. **Identify the integral form:** The integrand is a rational function with a quadratic denominator. We can use partial fraction decomposition if the denominator factors. 3. **Factor the denominator:** Solve $x^2 - x - 1 = 0$ using the quadratic formula: $$x = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}$$ So the factors are: $$x^2 - x - 1 = (x - \frac{1 + \sqrt{5}}{2})(x - \frac{1 - \sqrt{5}}{2})$$ 4. **Set up partial fractions:** $$\frac{2621}{x^2 - x - 1} = \frac{A}{x - \frac{1 + \sqrt{5}}{2}} + \frac{B}{x - \frac{1 - \sqrt{5}}{2}}$$ 5. **Find A and B:** Multiply both sides by the denominator: $$2621 = A\left(x - \frac{1 - \sqrt{5}}{2}\right) + B\left(x - \frac{1 + \sqrt{5}}{2}\right)$$ Set $x = \frac{1 + \sqrt{5}}{2}$: $$2621 = A\left(\frac{1 + \sqrt{5}}{2} - \frac{1 - \sqrt{5}}{2}\right) = A(\sqrt{5}) \Rightarrow A = \frac{2621}{\sqrt{5}}$$ Set $x = \frac{1 - \sqrt{5}}{2}$: $$2621 = B\left(\frac{1 - \sqrt{5}}{2} - \frac{1 + \sqrt{5}}{2}\right) = B(-\sqrt{5}) \Rightarrow B = -\frac{2621}{\sqrt{5}}$$ 6. **Rewrite the integral:** $$\int_1^{10} \frac{2621}{x^2 - x - 1} \, dx = \int_1^{10} \frac{2621/\sqrt{5}}{x - \frac{1 + \sqrt{5}}{2}} - \frac{2621/\sqrt{5}}{x - \frac{1 - \sqrt{5}}{2}} \, dx$$ 7. **Integrate term-by-term:** $$= \frac{2621}{\sqrt{5}} \int_1^{10} \frac{1}{x - \frac{1 + \sqrt{5}}{2}} \, dx - \frac{2621}{\sqrt{5}} \int_1^{10} \frac{1}{x - \frac{1 - \sqrt{5}}{2}} \, dx$$ 8. **Use the integral formula:** $$\int \frac{1}{x - a} dx = \ln|x - a| + C$$ 9. **Evaluate definite integrals:** $$= \frac{2621}{\sqrt{5}} \left[ \ln\left|x - \frac{1 + \sqrt{5}}{2}\right| \right]_1^{10} - \frac{2621}{\sqrt{5}} \left[ \ln\left|x - \frac{1 - \sqrt{5}}{2}\right| \right]_1^{10}$$ 10. **Calculate the values:** $$= \frac{2621}{\sqrt{5}} \left( \ln\left|10 - \frac{1 + \sqrt{5}}{2}\right| - \ln\left|1 - \frac{1 + \sqrt{5}}{2}\right| \right) - \frac{2621}{\sqrt{5}} \left( \ln\left|10 - \frac{1 - \sqrt{5}}{2}\right| - \ln\left|1 - \frac{1 - \sqrt{5}}{2}\right| \right)$$ 11. **Simplify using log properties:** $$= \frac{2621}{\sqrt{5}} \ln \frac{10 - \frac{1 + \sqrt{5}}{2}}{1 - \frac{1 + \sqrt{5}}{2}} - \frac{2621}{\sqrt{5}} \ln \frac{10 - \frac{1 - \sqrt{5}}{2}}{1 - \frac{1 - \sqrt{5}}{2}}$$ $$= \frac{2621}{\sqrt{5}} \ln \left( \frac{10 - \frac{1 + \sqrt{5}}{2}}{1 - \frac{1 + \sqrt{5}}{2}} \cdot \frac{1 - \frac{1 - \sqrt{5}}{2}}{10 - \frac{1 - \sqrt{5}}{2}} \right)$$ 12. **Final answer:** $$\boxed{\frac{2621}{\sqrt{5}} \ln \left( \frac{10 - \frac{1 + \sqrt{5}}{2}}{1 - \frac{1 + \sqrt{5}}{2}} \cdot \frac{1 - \frac{1 - \sqrt{5}}{2}}{10 - \frac{1 - \sqrt{5}}{2}} \right)}$$ This is the exact value of the definite integral.