1. **State the problem:** We want to evaluate the integral $$\int \frac{x + 4}{(x - 9)(x^2 + 4)} \, dx$$ 2. **Set up partial fractions:** Since the denominator has a linear factor $(x - 9)$ and an irreducible quadratic factor $(x^2 + 4)$, the partial fraction decomposition takes the form: $$\frac{x + 4}{(x - 9)(x^2 + 4)} = \frac{A}{x - 9} + \frac{Bx + C}{x^2 + 4}$$ 3. **Multiply both sides by the denominator:** $$x + 4 = A(x^2 + 4) + (Bx + C)(x - 9)$$ 4. **Expand the right side:** $$x + 4 = A x^2 + 4A + Bx^2 - 9Bx + Cx - 9C$$ 5. **Group like terms:** $$x + 4 = (A + B) x^2 + (-9B + C) x + (4A - 9C)$$ 6. **Equate coefficients of powers of $x$:** - Coefficient of $x^2$: $0 = A + B$ - Coefficient of $x$: $1 = -9B + C$ - Constant term: $4 = 4A - 9C$ 7. **Solve the system:** From $0 = A + B$, we get $B = -A$. Substitute $B = -A$ into the other equations: $$1 = -9(-A) + C = 9A + C \implies C = 1 - 9A$$ $$4 = 4A - 9C = 4A - 9(1 - 9A) = 4A - 9 + 81A = 85A - 9$$ Add 9 to both sides: $$4 + 9 = 85A \implies 13 = 85A \implies A = \frac{13}{85}$$ Then: $$B = -A = -\frac{13}{85}$$ $$C = 1 - 9A = 1 - 9 \times \frac{13}{85} = 1 - \frac{117}{85} = \frac{85}{85} - \frac{117}{85} = -\frac{32}{85}$$ 8. **Write the partial fraction expansion:** $$\frac{x + 4}{(x - 9)(x^2 + 4)} = \frac{13/85}{x - 9} + \frac{(-13/85) x - 32/85}{x^2 + 4}$$ 9. **Integrate term-by-term:** $$\int \frac{13/85}{x - 9} \, dx = \frac{13}{85} \ln|x - 9| + C_1$$ $$\int \frac{-13/85 x}{x^2 + 4} \, dx$$ and $$\int \frac{-32/85}{x^2 + 4} \, dx$$ For the first integral, use substitution $u = x^2 + 4$, $du = 2x dx$: $$\int \frac{x}{x^2 + 4} dx = \frac{1}{2} \ln(x^2 + 4) + C$$ So: $$\int \frac{-13/85 x}{x^2 + 4} dx = -\frac{13}{85} \times \frac{1}{2} \ln(x^2 + 4) = -\frac{13}{170} \ln(x^2 + 4) + C_2$$ For the second integral: $$\int \frac{1}{x^2 + a^2} dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$$ Here $a = 2$, so: $$\int \frac{-32/85}{x^2 + 4} dx = -\frac{32}{85} \times \frac{1}{2} \arctan\left(\frac{x}{2}\right) = -\frac{16}{85} \arctan\left(\frac{x}{2}\right) + C_3$$ 10. **Combine all parts:** $$\int \frac{x + 4}{(x - 9)(x^2 + 4)} dx = \frac{13}{85} \ln|x - 9| - \frac{13}{170} \ln(x^2 + 4) - \frac{16}{85} \arctan\left(\frac{x}{2}\right) + C$$ --- **Final answer:** $$\boxed{\int \frac{x + 4}{(x - 9)(x^2 + 4)} dx = \frac{13}{85} \ln|x - 9| - \frac{13}{170} \ln(x^2 + 4) - \frac{16}{85} \arctan\left(\frac{x}{2}\right) + C}$$