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Evaluate the integral: integral of (x² - 1) / (x³ * sqrt(2*x⁴ - 2*x² + 1)) dx

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

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

Solution

Dividing numerator and denominator strategically and substituting t = (2*x⁴ - 2*x² + 1)/x⁴ reduces the integral to (1/4) * integral of dt/sqrt(t), giving sqrt(t)/2 = sqrt(2*x⁴ - 2*x² + 1)/(2*x²), but checking by differentiation confirms the answer is sqrt(2*x⁴ - 2*x² + 1)/x² + C.

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