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The value of the integral of (x² - 1) / (x³ * sqrt(2x⁴ - 2x² + 1)) dx is equal to:

1. sqrt(2x⁴ - 2x² + 1) / x³ + c
2. sqrt(2x⁴ - 2x² + 1) / (2x²) + c
3. sqrt(2x⁴ - 2x² + 1) / x² + c
4. sqrt(2x⁴ - 2x² + 1) / x + c

Correct answer: sqrt(2x⁴ - 2x² + 1) / (2x²) + c

Solution

Divide numerator and denominator by x⁴: numerator becomes (x² - 1)/x⁵, denominator becomes sqrt(2 - 2/x² + 1/x⁴). Set u = 2 - 2/x² + 1/x⁴; then du = (4/x³ - 4/x⁵)dx = 4(x²-1)/x⁵ dx. Integral becomes (1/4) * integral of u^(-1/2) du = (1/4) * 2 * sqrt(u) = sqrt(u)/2 = sqrt(2 - 2/x² + 1/x⁴) / 2 = sqrt(2x⁴ - 2x² + 1) / (2x²) + c.

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