Indefinite Integration
Polynomial Integration
Grade 12
Question:
<p>The integral <span class="math">\int \frac{3x^{13} + 2x^{11}}{x^2}dx</span> is equal to (where <span class="math">C</span> is a constant of integration)</p>
<p>(a) <span class="math">\frac{x^4}{4} + \frac{x^{12}}{12} + C</span></p>
<p>(b) <span class="math">-\frac{x^4}{4} + \frac{x^{12}}{12} + C</span></p>
<p>(c) <span class="math">\frac{x^4}{4} + \frac{x^{12}}{9} + C</span></p>
<p>(d) <span class="math">\frac{x^4}{4} + \frac{x^{12}}{9} + C</span></p>
Step-by-Step Solution
Key Concept: Simplify rational expressions by dividing each term in the numerator by the denominator, then apply the power rule for integration.
<p><strong>Step 1:</strong> Simplify the integrand: <span class="math">\int \frac{3x^{13} + 2x^{11}}{x^2}dx = \int (3x^{11} + 2x^9)dx</span></p><p><strong>Step 2:</strong> Integrate term by term: <span class="math">= 3 \cdot \frac{x^{12}}{12} + 2 \cdot \frac{x^{10}}{10} + C = \frac{x^{12}}{4} + \frac{x^{10}}{5} + C</span></p><p>∴ Answer is (a).</p>
Correct Answer: a