1. **Problem:** Find constants $A$ and $B$ such that $$\frac{5x + 13}{(2x+1)(x+4)} \equiv \frac{A}{2x+1} + \frac{B}{x+4}$$ Then evaluate the integral $$\int_0^4 \frac{5x + 13}{(2x+1)(x+4)} \, dx$$ 2. **Partial Fraction Decomposition:** Multiply both sides by $(2x+1)(x+4)$: $$5x + 13 = A(x+4) + B(2x+1)$$ 3. **Expand and group terms:** $$5x + 13 = A x + 4A + 2Bx + B$$ $$5x + 13 = (A + 2B)x + (4A + B)$$ 4. **Equate coefficients:** For $x$: $$5 = A + 2B$$ For constants: $$13 = 4A + B$$ 5. **Solve the system:** From $5 = A + 2B$, express $A = 5 - 2B$ Substitute into $13 = 4A + B$: $$13 = 4(5 - 2B) + B = 20 - 8B + B = 20 - 7B$$ $$7B = 20 - 13 = 7 \implies B = 1$$ Then, $$A = 5 - 2(1) = 3$$ 6. **Rewrite the integrand:** $$\frac{5x + 13}{(2x+1)(x+4)} = \frac{3}{2x+1} + \frac{1}{x+4}$$ 7. **Integral evaluation:** $$\int_0^4 \frac{5x + 13}{(2x+1)(x+4)} \, dx = \int_0^4 \left( \frac{3}{2x+1} + \frac{1}{x+4} \right) dx$$ 8. **Integrate each term:** $$\int \frac{3}{2x+1} dx = 3 \int \frac{1}{2x+1} dx = 3 \cdot \frac{1}{2} \ln|2x+1| = \frac{3}{2} \ln|2x+1|$$ $$\int \frac{1}{x+4} dx = \ln|x+4|$$ 9. **Evaluate definite integral:** $$\int_0^4 \frac{5x + 13}{(2x+1)(x+4)} dx = \left[ \frac{3}{2} \ln|2x+1| + \ln|x+4| \right]_0^4$$ Calculate at $x=4$: $$\frac{3}{2} \ln(2(4)+1) + \ln(4+4) = \frac{3}{2} \ln(9) + \ln(8)$$ At $x=0$: $$\frac{3}{2} \ln(1) + \ln(4) = 0 + \ln(4)$$ 10. **Subtract:** $$\left( \frac{3}{2} \ln(9) + \ln(8) \right) - \ln(4) = \frac{3}{2} \ln(9) + \ln\left( \frac{8}{4} \right) = \frac{3}{2} \ln(9) + \ln(2)$$ 11. **Simplify logarithms:** $$\ln(9) = \ln(3^2) = 2 \ln(3)$$ So, $$\frac{3}{2} \ln(9) = \frac{3}{2} \times 2 \ln(3) = 3 \ln(3)$$ Final answer: $$3 \ln(3) + \ln(2) = \ln(3^3) + \ln(2) = \ln(27) + \ln(2) = \ln(54)$$ **Therefore,** $$\int_0^4 \frac{5x + 13}{(2x+1)(x+4)} dx = \ln(54)$$