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Evaluate the indefinite integral ∫ (2x¹²+5x⁹)/((x⁵+x³+1)³) dx. It is equal to:

1. (x⁵)/(2(x⁵+x³+1)²)+C
2. -(x¹⁰)/(2(x⁵+x³+1)²)+C
3. -(x⁵)/((x⁵+x³+1)²)+C
4. (x¹⁰)/(2(x⁵+x³+1)²)+C

Correct answer: (x¹⁰)/(2(x⁵+x³+1)²)+C

Solution

Dividing numerator and denominator by x^15 and letting t=(x^5+x^3+1)/x^5 gives dt=-(2x^-3+5x^-6)dx, so the integral = ∫ -dt/t^3 = 1/(2t^2) = x^10/(2(x^5+x^3+1)^2) + C.

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