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Evaluate the integral ∫ [sin² x cos² x] / [(sin⁵ x + cos⁵ x sin² x + sin³ x cos² x + cos⁵ x)²] dx, where C denotes the constant of integration.

1. −1 / [3(1 + tan³ x)] + C
2. 1 / (1 + cot³ x) + C
3. −1 / (1 + cot³ x) + C
4. 1 / [3(1 + tan³ x)] + C

Correct answer: −1 / [3(1 + tan³ x)] + C

Solution

With denominator (sin^3 x+cos^3 x)^2, d/dx[-1/(3(1+tan^3 x))] = tan^2 x sec^2 x/(1+tan^3 x)^2 = sin^2 x cos^2 x/(sin^3 x+cos^3 x)^2, which is the integrand. Hence the integral equals -1/[3(1+tan^3 x)] + C.

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