Question

Evaluate the integral: \[ \int \frac{6x^3 + 9x^2}{x^4 + 3x^3 - 9x^2} dx. \]

• A. \( 3x \log |x^2 + 3x - 9| + C \)
• B. \( 6x \log |x^2 + 3x - 9| + C \)
• C. \( 6 \log |x^2 + 3x - 9| + C \)
• D. \( x \log |x^2 + 3x - 9| + C \)
• E. \( 3 \log |x^2 + 3x - 9| + C \)

Hint:

For rational functions, try expressing the numerator as a multiple of the derivative of the denominator.

Answer 1

Answer: (E)

Step 1: Recognizing the pattern The denominator is \( x^4 + 3x^3 - 9x^2 \), and its derivative is: \[ \frac{d}{dx} (x^2 + 3x - 9) = 2x + 3. \] Step 2: Factor the numerator Rewriting: \[ 6x^3 + 9x^2 = 3(2x^3 + 3x^2). \] Factoring: \[ = 3x^2(2x + 3). \] Step 3: Express the integral \[ \int \frac{6x^3 + 9x^2}{x^4 + 3x^3 - 9x^2} dx = 3 \int \frac{2x + 3}{x^2 + 3x - 9} dx. \] Step 4: Solve using logarithmic integration \[ 3 \int \frac{d(x^2 + 3x - 9)}{x^2 + 3x - 9}. \] \[ = 3 \log |x^2 + 3x - 9| + C. \] Thus, the correct answer is (E) \( 3 \log |x^2 + 3x - 9| + C \).