Step-by-Step Solution
Key Concept: Factor the denominator as x²(x² + 1) and use partial fraction decomposition: 1/(x²(x² + 1)) = A/x + B/x² + (Cx + D)/(x² + 1). The algebraic manipulation reveals A = 0, B = 1, C = 0, D = -1, allowing separation into integrable terms.
<p><strong>Step 1:</strong> Factor the denominator: x⁴ + x² = x²(x² + 1)</p><p><strong>Step 2:</strong> Set up partial fraction decomposition:</p><p>1/(x²(x² + 1)) = A/x + B/x² + (Cx + D)/(x² + 1)</p><p><strong>Step 3:</strong> Multiply both sides by x²(x² + 1):</p><p>1 = Ax(x² + 1) + B(x² + 1) + (Cx + D)x²</p><p><strong>Step 4:</strong> Solve for coefficients:</p><p>• x = 0: 1 = B(1) → B = 1</p><p>• Coefficient of x³: 0 = A + C → C = -A</p><p>• Coefficient of x²: 0 = B + D → D = -1</p><p>• Coefficient of x: 0 = A → A = 0, C = 0</p><p><strong>Step 5:</strong> Integrate:</p><p>6∫[1/x² - 1/(x² + 1)]dx = 6[-1/x - arctan(x)] + C</p><p><strong>Step 6:</strong> Simplify:</p><p>∴ Answer: <strong>-6/x - 6arctan(x) + C</strong> (or -6/x - 6tan⁻¹(x) + C)</p>
Correct Answer: -6