1. **State the problem:** Express the rational function $$\frac{19 - x}{(x - 7)(x + 4)}$$ as the sum of its partial fractions and then find the integral $$\int \frac{19 - x}{(x - 7)(x + 4)} \, dx$$. 2. **Set up the partial fractions:** We want to write $$\frac{19 - x}{(x - 7)(x + 4)} = \frac{A}{x - 7} + \frac{B}{x + 4}$$ where $$A$$ and $$B$$ are constants to be determined. 3. **Multiply both sides by the denominator:** $$19 - x = A(x + 4) + B(x - 7)$$ 4. **Expand the right side:** $$19 - x = A x + 4A + B x - 7B = (A + B) x + (4A - 7B)$$ 5. **Equate coefficients:** For the $$x$$ terms: $$-1 = A + B$$ For the constant terms: $$19 = 4A - 7B$$ 6. **Solve the system:** From $$-1 = A + B$$, we get $$A = -1 - B$$. Substitute into the second equation: $$19 = 4(-1 - B) - 7B = -4 - 4B - 7B = -4 - 11B$$ Add 4 to both sides: $$23 = -11B$$ So, $$B = -\frac{23}{11}$$ Then, $$A = -1 - \left(-\frac{23}{11}\right) = -1 + \frac{23}{11} = \frac{12}{11}$$ 7. **Rewrite the partial fractions:** $$\frac{19 - x}{(x - 7)(x + 4)} = \frac{12/11}{x - 7} - \frac{23/11}{x + 4}$$ 8. **Integrate term-by-term:** $$\int \frac{19 - x}{(x - 7)(x + 4)} \, dx = \int \frac{12/11}{x - 7} \, dx - \int \frac{23/11}{x + 4} \, dx$$ 9. **Use the integral formula:** $$\int \frac{1}{x - a} \, dx = \ln|x - a| + C$$ 10. **Final integral:** $$= \frac{12}{11} \ln|x - 7| - \frac{23}{11} \ln|x + 4| + C$$ **Answer:** $$A = \frac{12}{11}, \quad B = -\frac{23}{11}$$ $$\int \frac{19 - x}{(x - 7)(x + 4)} \, dx = \frac{12}{11} \ln|x - 7| - \frac{23}{11} \ln|x + 4| + C$$