Question

Evaluate \( \int \frac{x^2}{(x+1)^2(x+2)^2} \, dx \)

• A. \( \log|x + 1| - \frac{4}{x + 2} + c \)
• B. \( \log|x + 1| - \frac{4}{x + 2} + \frac{3}{(x + 2)^2} + c \)
• C. \( \log|x + 1| + \frac{1}{x + 2} + c \)
• D. \( \log|x + 1| - \frac{4}{x + 2} - \frac{3}{(x + 2)^2} + c \)

Hint:

When solving integrals involving rational functions, use partial fraction decomposition to break the integrand into simpler terms.

Answer 1

The Correct Option is A

Step 1: Break down the integrand.
We want to perform partial fraction decomposition on \[ \frac{x^2}{(x+1)^2(x+2)^2} \] Step 2: Set up partial fractions.
\[ \frac{x^2}{(x+1)^2(x+2)^2} = \frac{A}{x+1} + \frac{B}{(x+1)^2} + \frac{C}{x+2} + \frac{D}{(x+2)^2} \] Step 3: Solve for the constants.
Multiply both sides by \( (x+1)^2(x+2)^2 \) and solve for \( A, B, C, D \). After simplification, we get \[ A = 1, B = -4, C = 0, D = 0 \] Step 4: Integrate each term.
\[ \int \frac{1}{x+1} \, dx = \log|x+1| \] \[ \int \frac{-4}{(x+2)} \, dx = -\frac{4}{x+2} \] Step 5: Combine the results.
\[ \int \frac{x^2}{(x+1)^2(x+2)^2} \, dx = \log|x + 1| - \frac{4}{x + 2} + c \] Step 6: Conclusion.
The final answer is \( \log|x + 1| - \frac{4}{x + 2} + c \).