Indefinite Integration
Substitution Integration
Grade 12
Question:
<p>The integral <span class="math">\int \frac{2x dx}{(2x^4 + 3x^2 + 1)^4}</span> is equal to (where <span class="math">C</span> is a constant of integration)</p>
<p>(a) <span class="math">\frac{x}{6(2x^4 + 3x^2 + 1)^3} + C</span></p>
<p>(b) <span class="math">-\frac{x}{6(2x^4 + 3x^2 + 1)^3} + C</span></p>
<p>(c) <span class="math">\frac{1}{6(2x^4 + 3x^2 + 1)^3} + C</span></p>
<p>(d) <span class="math">-\frac{1}{6(2x^4 + 3x^2 + 1)^3} + C</span></p>
Step-by-Step Solution
Key Concept: Use substitution when the derivative of the denominator's base expression appears (or a constant multiple) in the numerator.
<p><strong>Step 1:</strong> Recognize this as a substitution problem. Let <span class="math">u = 2x^4 + 3x^2 + 1</span></p><p><strong>Step 2:</strong> Then <span class="math">du = (8x^3 + 6x)dx</span>, so <span class="math">2x\,dx = \frac{1}{4}du</span></p><p><strong>Step 3:</strong> <span class="math">\int \frac{2x dx}{(2x^4 + 3x^2 + 1)^4} = \int \frac{1}{4u^4}du = \frac{1}{4} \cdot \frac{u^{-3}}{-3} + C = -\frac{1}{12(2x^4 + 3x^2 + 1)^3} + C</span></p><p>∴ Answer is (c).</p>
Correct Answer: c