1. **State the problem:** Evaluate the definite integral $$\int_2^4 \frac{x^2 + 2x + 4}{x^3 + 4x} \, dx.$$\n\n2. **Simplify the integrand:** Factor the denominator:\n$$x^3 + 4x = x(x^2 + 4).$$\nSo the integral becomes:\n$$\int_2^4 \frac{x^2 + 2x + 4}{x(x^2 + 4)} \, dx.$$\n\n3. **Use partial fraction decomposition:** Write\n$$\frac{x^2 + 2x + 4}{x(x^2 + 4)} = \frac{A}{x} + \frac{Bx + C}{x^2 + 4}.$$\nMultiply both sides by the denominator:\n$$x^2 + 2x + 4 = A(x^2 + 4) + (Bx + C)x = A x^2 + 4A + B x^2 + C x.$$\nGroup terms:\n$$x^2 + 2x + 4 = (A + B) x^2 + C x + 4A.$$\n\n4. **Equate coefficients:**\n- Coefficient of $x^2$: $1 = A + B$\n- Coefficient of $x$: $2 = C$\n- Constant term: $4 = 4A$\n\nFrom $4 = 4A$, we get $A = 1$.\nFrom $1 = A + B$, we get $1 = 1 + B \Rightarrow B = 0$.\nFrom $2 = C$, we get $C = 2$.\n\n5. **Rewrite the integrand:**\n$$\frac{x^2 + 2x + 4}{x(x^2 + 4)} = \frac{1}{x} + \frac{2}{x^2 + 4}.$$\n\n6. **Integrate term-by-term:**\n$$\int_2^4 \left( \frac{1}{x} + \frac{2}{x^2 + 4} \right) dx = \int_2^4 \frac{1}{x} dx + \int_2^4 \frac{2}{x^2 + 4} dx.$$\n\n7. **Integrate each part:**\n- $$\int \frac{1}{x} dx = \ln|x| + C.$$\n- $$\int \frac{2}{x^2 + 4} dx = 2 \int \frac{1}{x^2 + 2^2} dx = 2 \cdot \frac{1}{2} \arctan \frac{x}{2} + C = \arctan \frac{x}{2} + C.$$\n\n8. **Evaluate definite integrals:**\n$$\int_2^4 \frac{1}{x} dx = \ln 4 - \ln 2 = \ln \frac{4}{2} = \ln 2,$$\n$$\int_2^4 \frac{2}{x^2 + 4} dx = \arctan 2 - \arctan 1.$$\n\n9. **Final answer:**\n$$\int_2^4 \frac{x^2 + 2x + 4}{x^3 + 4x} dx = \ln 2 + \arctan 2 - \arctan 1.$$