<p>If \(f(x) = \displaystyle\int \dfrac{(3x^4 - 1)}{(x^4 + x + 1)^2}\, dx\) and \(f(0) = 0\), then \(f(-1)\) is equal to:</p>
Step-by-Step Solution
Key Concept: Recognize that the numerator (3x⁴ - 1) can be decomposed as a linear combination of (x⁴ + x + 1) and its derivative (4x³ + 1), enabling substitution u = x⁴ + x + 1.
<p><strong>Step 1:</strong> Decompose the numerator as: 3x⁴ - 1 = 3(x⁴ + x + 1) - 3x - 4</p><p><strong>Step 2:</strong> Split the integral: ∫[3(x⁴ + x + 1) - 3x - 4]/(x⁴ + x + 1)² dx = 3∫1/(x⁴ + x + 1) dx - ∫(3x + 4)/(x⁴ + x + 1)² dx</p><p><strong>Step 3:</strong> For the second integral, recognize that d/dx(x⁴ + x + 1) = 4x³ + 1. Rewrite: ∫(3x + 4)/(x⁴ + x + 1)² dx using substitution u = x⁴ + x + 1, noting that the numerator relates to the derivative.</p><p><strong>Step 4:</strong> After careful analysis, the first integral is complex. Instead, observe: d/dx[x/(x⁴ + x + 1)] = [(x⁴ + x + 1) - x(4x³ + 1)]/(x⁴ + x + 1)² = (x⁴ + x + 1 - 4x⁴ - x)/(x⁴ + x + 1)² = (-3x⁴ + 1)/(x⁴ + x + 1)²</p><p><strong>Step 5:</strong> Therefore: ∫(3x⁴ - 1)/(x⁴ + x + 1)² dx = -x/(x⁴ + x + 1) + C</p><p><strong>Step 6:</strong> Using f(0) = 0: 0 = -0/(1) + C, so C = 0</p><p><strong>Step 7:</strong> Thus f(x) = -x/(x⁴ + x + 1)</p><p><strong>Step 8:</strong> f(-1) = -(-1)/(1 - 1 + 1) = 1/1 = 1</p><p>∴ Answer: A</p>
Correct Answer: A