1. **Problem statement:** Find constants $A$, $B$, and $C$ such that $$\frac{x^2 + 8x - 3}{x + 2} \equiv Ax + B + \frac{C}{x + 2}, \quad x \in \mathbb{R}, x \neq -2$$ Then, use this to evaluate the integral $$\int_0^6 \frac{x^2 + 8x - 3}{x + 2} \, dx$$ 2. **Step 1: Express the given rational function as a sum of a polynomial and a proper fraction.** We want to write $$\frac{x^2 + 8x - 3}{x + 2} = Ax + B + \frac{C}{x + 2}$$ Multiply both sides by $x + 2$: $$x^2 + 8x - 3 = (Ax + B)(x + 2) + C$$ 3. **Step 2: Expand the right side:** $$(Ax + B)(x + 2) + C = A x^2 + 2 A x + B x + 2 B + C = A x^2 + (2A + B) x + (2B + C)$$ 4. **Step 3: Equate coefficients of powers of $x$ on both sides:** From the left side: $x^2$ coefficient is 1, $x$ coefficient is 8, constant term is $-3$. From the right side: - Coefficient of $x^2$ is $A$ - Coefficient of $x$ is $2A + B$ - Constant term is $2B + C$ So, $$A = 1$$ $$2A + B = 8$$ $$2B + C = -3$$ 5. **Step 4: Solve for $A$, $B$, and $C$:** Since $A=1$, substitute into second equation: $$2(1) + B = 8 \implies 2 + B = 8 \implies B = 6$$ Substitute $B=6$ into third equation: $$2(6) + C = -3 \implies 12 + C = -3 \implies C = -15$$ 6. **Step 5: Write the decomposition:** $$\frac{x^2 + 8x - 3}{x + 2} = x + 6 - \frac{15}{x + 2}$$ 7. **Step 6: Set up the integral using the decomposition:** $$\int_0^6 \frac{x^2 + 8x - 3}{x + 2} \, dx = \int_0^6 \left(x + 6 - \frac{15}{x + 2}\right) dx$$ 8. **Step 7: Integrate term-by-term:** $$\int_0^6 x \, dx = \left[ \frac{x^2}{2} \right]_0^6 = \frac{36}{2} = 18$$ $$\int_0^6 6 \, dx = \left[6x\right]_0^6 = 36$$ $$\int_0^6 \frac{15}{x + 2} \, dx = 15 \int_0^6 \frac{1}{x + 2} \, dx = 15 \left[ \ln|x + 2| \right]_0^6 = 15 (\ln 8 - \ln 2) = 15 \ln 4$$ 9. **Step 8: Combine the results:** $$\int_0^6 \frac{x^2 + 8x - 3}{x + 2} \, dx = 18 + 36 - 15 \ln 4 = 54 - 15 \ln 4$$ 10. **Step 9: Express $\ln 4$ in terms of $\ln 2$:** $$\ln 4 = \ln (2^2) = 2 \ln 2$$ So, $$54 - 15 \ln 4 = 54 - 15 (2 \ln 2) = 54 - 30 \ln 2$$ **Final answers:** - $A = 1$, $B = 6$, $C = -15$ - $$\int_0^6 \frac{x^2 + 8x - 3}{x + 2} \, dx = 54 - 30 \ln 2$$