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Let f'(x) = (5x³ - 6) / sqrt(x⁷ - 3x⁴ + 4x²) and f(2) = sqrt(6). Find the value of the integral from 0 to 2 of x * f(x) dx.

1. (A) 4*sqrt(6) - 6
2. (B) 2*sqrt(6) - 2
3. (C) 6*sqrt(6) - 8
4. (D) 3*sqrt(6) - 4

Correct answer: (A) 4*sqrt(6) - 6

Solution

Integration by parts on integral x*f(x) dx transforms it into a simpler integral. The key step is recognising that the numerator 5x³ - 6 is related to the derivative of x⁵ - 3x² + 4, enabling a substitution.

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