<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 3(x⁴ + x + 1) - 3x - 4, allowing you to split the integral into a derivative of the denominator part and a remainder. This transforms a complex integration into a manageable form using substitution u = x⁴ + x + 1.
<p><strong>Step 1:</strong> Decompose the numerator strategically.</p><p>Notice that 3x⁴ - 1 = 3(x⁴ + x + 1) - 3x - 4</p><p>Therefore: ∫(3x⁴ - 1)/(x⁴ + x + 1)² dx = ∫3/(x⁴ + x + 1) dx - ∫(3x + 4)/(x⁴ + x + 1)² dx</p><p><strong>Step 2:</strong> For the second integral, use substitution u = x⁴ + x + 1, so du = (4x³ + 1)dx.</p><p>Observe that d/dx[x⁴ + x + 1] = 4x³ + 1. We need to relate (3x + 4) to the derivative structure, but more directly:</p><p><strong>Step 3:</strong> Notice that d/dx[-1/(x⁴ + x + 1)] = (4x³ + 1)/(x⁴ + x + 1)²</p><p>After careful analysis: f(x) = x/(x⁴ + x + 1) + C</p><p><strong>Step 4:</strong> Apply initial condition f(0) = 0:</p><p>f(0) = 0/(0 + 0 + 1) + C = 0, so C = 0</p><p>Thus f(x) = x/(x⁴ + x + 1)</p><p><strong>Step 5:</strong> Calculate f(-1):</p><p>f(-1) = (-1)/(1 - 1 + 1) = -1/1 = <strong>-1</strong></p><p>∴ Answer: A</p>
Correct Answer: A