Question:
Evaluate the integral:
\[
\int \frac{6x^3 + 9x^2}{x^4 + 3x^3 - 9x^2} dx.
\]
Evaluate the integral:
\[
\int \frac{6x^3 + 9x^2}{x^4 + 3x^3 - 9x^2} dx.
\]
Show Hint
For rational functions, try expressing the numerator as a multiple of the derivative of the denominator.
Updated On: Apr 7, 2026
- \( 3x \log |x^2 + 3x - 9| + C \)
- \( 6x \log |x^2 + 3x - 9| + C \)
- \( 6 \log |x^2 + 3x - 9| + C \)
- \( x \log |x^2 + 3x - 9| + C \)
- \( 3 \log |x^2 + 3x - 9| + C \)
Show Solution
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The Correct Option is
Solution and Explanation
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) \( 3 \log |x^2 + 3x - 9| + C \).
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