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Evaluate the integral ∫ (dx)/(x² (x⁴+1)^(3/4)):

1. -(x⁴+1)^(1/4)+c
2. - ((x⁴+1)/(x⁴))^(1/4)+c
3. ((x⁴+1)/(x⁴))^(1/4)+c
4. (x⁴+1)^(1/4)+c

Correct answer: - ((x⁴+1)/(x⁴))^(1/4)+c

Solution

Factor x^4: integrand = 1/(x^5 (1+x^-4)^(3/4)). Let u=1+x^-4, du=-4x^-5 dx, giving -(1/4) integral u^(-3/4) du = -u^(1/4)+c = -((x^4+1)/x^4)^(1/4)+c.

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