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. **Partial fraction decomposition formula:** For distinct linear factors, we 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 of like terms:** - Coefficient of $x$: $-1 = A + B$ - Constant term: $19 = 4A - 7B$ 6. **Solve the system:** From $-1 = A + B$, we get $A = -1 - B$. Substitute into $19 = 4A - 7B$: $$19 = 4(-1 - B) - 7B = -4 - 4B - 7B = -4 - 11B$$ 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. **Evaluate the integrals:** $$= \frac{12}{11} \ln|x - 7| - \frac{23}{11} \ln|x + 4| + C$$ **Final answer:** $$\int \frac{19 - x}{(x - 7)(x + 4)} \, dx = \frac{12}{11} \ln|x - 7| - \frac{23}{11} \ln|x + 4| + C$$