Indefinite Integration
Integration of Rational Functions
Grade 12
Question:
<p>Find <span>\(\int \frac{3x+1}{(x+1)^3} dx\)</span></p>
<p>(A) <span>\(\frac{1}{4}\log|x+1| - \frac{1}{4}\log|x-1| - \frac{1}{2(x+1)} + c\)</span></p>
<p>(B) <span>\(\frac{1}{4}\log|x-1| - \frac{1}{4}\log|x+1| - \frac{1}{2(x+1)} - \frac{1}{2(x+1)^2} + c\)</span></p>
<p>(C) Other form</p>
<p>(D) None of these</p>
Step-by-Step Solution
Key Concept: Use partial fractions or algebraic manipulation to simplify the rational function before integration.
<p><strong>Step 1:</strong> Decompose using partial fractions or integration by parts.<br/><strong>Step 2:</strong> Integrate each term separately.<br/><strong>Step 3:</strong> Combine the results to get the final answer.</p>
Correct Answer: B