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The integral ∫[π/6 to π/4] dx / [sin 2x (tan⁵ x + cot⁵ x)] equals: (1) π/40 (2) (1/20) tan⁻¹(1/(9√3)) (3) (1/10) [π/4 - tan⁻¹(1/(9√3))] (4) (1/5) [π/4 - tan⁻¹(1/(3√3))]

1. π/40
2. (1/20) tan⁻¹(1/(9√3))
3. (1/10) [π/4 - tan⁻¹(1/(9√3))]
4. (1/5) [π/4 - tan⁻¹(1/(3√3))]

Correct answer: (1/10) [π/4 - tan⁻¹(1/(9√3))]

Solution

The correct option is derived from evaluating the integral using trigonometric identities and substitutions, which simplify the expression effectively, leading to the result that matches the form of option C.

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