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Evaluate the integral ∫ [sin^2 x cos^2 x] / [(sin^5 x + cos^5 x sin^2 x + sin^3 x cos^2 x + cos^5 x)^2] dx, where C denotes the constant of integration.

1. −1 / [3(1 + tan^3 x)] + C
2. 1 / (1 + cot^3 x) + C
3. −1 / (1 + cot^3 x) + C
4. 1 / [3(1 + tan^3 x)] + C

Correct answer: 1 / [3(1 + tan^3 x)] + C

Solution

The correct option is derived from the integral's structure, which simplifies to a form that matches the antiderivative involving the function 1/(1 + tan^3 x). The presence of tan^3 x in the denominator indicates that the integral evaluates to a function that decreases as tan x increases, leading to the specific form of the answer.

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