1. **Problem statement:** Express the rational function \(\frac{21-x}{(x-5)(x+4)}\) as the sum of its partial fractions of the form \(\frac{A}{x-5} + \frac{B}{x+4}\), and then find the integral \(\int \frac{21-x}{(x-5)(x+4)} \, dx\). 2. **Express as partial fractions:** Assume \(\frac{21-x}{(x-5)(x+4)} = \frac{A}{x-5} + \frac{B}{x+4}\). 3. Multiply both sides by the denominator \((x-5)(x+4)\) to clear fractions: $$21 - x = A(x + 4) + B(x - 5)$$ 4. Expand the right side: $$21 - x = A x + 4A + B x - 5 B = (A + B)x + (4A - 5B)$$ 5. Equate coefficients of like terms: For \(x\): $$-1 = A + B$$ For constant term: $$21 = 4A - 5B$$ 6. Solve the system: From \(A + B = -1\) we get \(A = -1 - B\). Substitute into second equation: $$21 = 4(-1 - B) - 5B = -4 - 4B - 5B = -4 - 9B$$ Rearranged: $$21 + 4 = -9B \,\Rightarrow\, 25 = -9B \,\Rightarrow\, B = -\frac{25}{9}$$ Then: $$A = -1 - \left(-\frac{25}{9}\right) = -1 + \frac{25}{9} = \frac{25}{9} - \frac{9}{9} = \frac{16}{9}$$ 7. **Partial fraction form:** $$\frac{21 - x}{(x-5)(x+4)} = \frac{16/9}{x-5} - \frac{25/9}{x+4}$$ 8. **Integrate:** $$\int \frac{21-x}{(x-5)(x+4)} \, dx = \int \left(\frac{16/9}{x-5} - \frac{25/9}{x+4} \right) \, dx = \frac{16}{9} \int \frac{1}{x-5} \, dx - \frac{25}{9} \int \frac{1}{x+4} \, dx$$ 9. Compute the integrals: $$= \frac{16}{9} \ln|x-5| - \frac{25}{9} \ln|x+4| + C$$ where \(C\) is the constant of integration. **Final answer:** $$\int \frac{21-x}{(x-5)(x+4)} \, dx = \frac{16}{9} \ln|x-5| - \frac{25}{9} \ln|x+4| + C$$