Indefinite Integration
Rational Function Integration
Grade 12
Question:
<p>The integral <span class="math">\int \frac{(x+1)^3 + x + x^2 dx}{1 + x}</span> is equal to</p>
<p>(a) <span class="math">\frac{x^5}{5} + \frac{x^3}{3} + C</span></p>
<p>(b) <span class="math">\frac{x^5}{5} + \frac{x^3}{3} + C</span></p>
<p>(c) <span class="math">\frac{x^5}{5} + \frac{x^{10}}{10} + C</span></p>
<p>(d) <span class="math">-\frac{x^5}{5} + \frac{x^{10}}{10} + C</span></p>
Step-by-Step Solution
Key Concept: When the numerator has degree greater than or equal to the denominator, perform polynomial division before integrating.
<p><strong>Step 1:</strong> Expand and simplify the integrand by polynomial division or algebraic manipulation.</p><p><strong>Step 2:</strong> Integrate the resulting polynomial expression.</p><p>∴ Answer is (a).</p>
Correct Answer: a