We aim to compute the integral of $\frac{x^2 + 2}{x^3 + 4x^2} \, dx$. Let’s proceed step by step:

Step 1: Simplify the integrand

Factorize the denominator: $x^3 + 4x^2 = x^2(x + 4).$ Thus, the integral becomes: $\int \frac{x^2 + 2}{x^2(x + 4)} \, dx.$

Step 2: Separate the terms

Split the fraction into partial fractions: $\frac{x^2 + 2}{x^2(x + 4)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x + 4}.$

Multiply through by $x^2(x + 4)$ to find $A$, $B$, and $C$: $x^2 + 2 = A x(x + 4) + B(x + 4) + Cx^2.$

Expanding this: $x^2 + 2 = A x^2 + 4A x + Bx + 4B + Cx^2.$

Combine like terms: $x^2 + 2 = (A + C)x^2 + (4A + B)x + 4B.$

Equate coefficients:

1. $A + C = 1$ (coefficient of $x^2$),
2. $4A + B = 0$ (coefficient of $x$),
3. $4B = 2$ (constant term).

From $4B = 2$: $B = \frac{1}{2}.$

From $4A + B = 0$: $4A + \frac{1}{2} = 0 \quad \implies \quad A = -\frac{1}{8}.$

From $A + C = 1$: $-\frac{1}{8} + C = 1 \quad \implies \quad C = \frac{9}{8}.$

Thus, the partial fraction decomposition is: $\frac{x^2 + 2}{x^2(x + 4)} = \frac{-\frac{1}{8}}{x} + \frac{\frac{1}{2}}{x^2} + \frac{\frac{9}{8}}{x + 4}.$

Step 3: Integrate each term

The integral becomes: $\int \frac{x^2 + 2}{x^2(x + 4)} \, dx = \int \frac{-\frac{1}{8}}{x} \, dx + \int \frac{\frac{1}{2}}{x^2} \, dx + \int \frac{\frac{9}{8}}{x + 4} \, dx.$

1. First term: $\int \frac{-\frac{1}{8}}{x} \, dx = -\frac{1}{8} \ln|x|.$

2. Second term: $\int \frac{\frac{1}{2}}{x^2} \, dx = \frac{1}{2} \int x^{-2} \, dx = \frac{1}{2} \left( -x^{-1} \right) = -\frac{1}{2x}.$

3. Third term: $\int \frac{\frac{9}{8}}{x + 4} \, dx = \frac{9}{8} \ln|x + 4|.$

Step 4: Combine the results

The final integral is: $\int \frac{x^2 + 2}{x^3 + 4x^2} \, dx = -\frac{1}{8} \ln|x| - \frac{1}{2x} + \frac{9}{8} \ln|x + 4| + C,$ where $C$ is the constant of integration.

Let me know if you have questions or want any specific steps clarified. 😊

Follow-up Questions

1. Would you like a more detailed explanation of the partial fraction decomposition process?
2. Do you need help with integrating similar rational functions?
3. Would you like me to explain the rules for solving definite integrals involving logarithms?
4. Should I verify this solution with numerical integration?
5. Would you like to see a graphical representation of the integrand?

Tip

Always check if the denominator can be factorized to simplify rational integrals using partial fractions!