1. **State the problem:** We want to express the integrand $$\frac{48x^2}{(x-18)(x+6)^2}$$ as a sum of partial fractions and then evaluate the integral. 2. **Set up the partial fraction decomposition:** Since the denominator has a linear factor $(x-18)$ and a repeated linear factor $(x+6)^2$, the decomposition is: $$\frac{48x^2}{(x-18)(x+6)^2} = \frac{A}{x-18} + \frac{B}{x+6} + \frac{C}{(x+6)^2}$$ 3. **Multiply both sides by the denominator:** $$48x^2 = A(x+6)^2 + B(x-18)(x+6) + C(x-18)$$ 4. **Expand terms:** - $(x+6)^2 = x^2 + 12x + 36$ - $(x-18)(x+6) = x^2 - 12x - 108$ So, $$48x^2 = A(x^2 + 12x + 36) + B(x^2 - 12x - 108) + C(x - 18)$$ 5. **Group like terms:** $$48x^2 = (A + B)x^2 + (12A - 12B + C)x + (36A - 108B - 18C)$$ 6. **Equate coefficients:** - Coefficient of $x^2$: $48 = A + B$ - Coefficient of $x$: $0 = 12A - 12B + C$ - Constant term: $0 = 36A - 108B - 18C$ 7. **Solve the system:** From $48 = A + B$, we get $B = 48 - A$. Substitute into the $x$ coefficient equation: $$0 = 12A - 12(48 - A) + C = 12A - 576 + 12A + C = 24A - 576 + C$$ So, $$C = 576 - 24A$$ Substitute $B$ and $C$ into the constant term equation: $$0 = 36A - 108(48 - A) - 18(576 - 24A)$$ $$0 = 36A - 5184 + 108A - 10368 + 432A$$ $$0 = (36 + 108 + 432)A - (5184 + 10368)$$ $$0 = 576A - 15552$$ Solve for $A$: $$A = \frac{15552}{576} = 27$$ Then, $$B = 48 - 27 = 21$$ $$C = 576 - 24 \times 27 = 576 - 648 = -72$$ 8. **Rewrite the integrand:** $$\frac{48x^2}{(x-18)(x+6)^2} = \frac{27}{x-18} + \frac{21}{x+6} - \frac{72}{(x+6)^2}$$ 9. **Integrate term-by-term:** $$\int \frac{48x^2}{(x-18)(x+6)^2} dx = \int \frac{27}{x-18} dx + \int \frac{21}{x+6} dx - \int \frac{72}{(x+6)^2} dx$$ 10. **Evaluate each integral:** - $$\int \frac{27}{x-18} dx = 27 \ln|x-18| + C_1$$ - $$\int \frac{21}{x+6} dx = 21 \ln|x+6| + C_2$$ - $$\int \frac{72}{(x+6)^2} dx = 72 \int (x+6)^{-2} dx = 72 \left(-\frac{1}{x+6}\right) + C_3 = -\frac{72}{x+6} + C_3$$ 11. **Combine results:** $$\int \frac{48x^2}{(x-18)(x+6)^2} dx = 27 \ln|x-18| + 21 \ln|x+6| - \frac{72}{x+6} + C$$ where $C$ is the constant of integration.