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Let Iₙ = ∫ tanⁿ x dx, where n > 1. If I₄ + I₆ = a tan⁵ x + b x⁵ + C, with C as the constant of integration, then the ordered pair (a, b) is:

1. (−1/5, 0)
2. (−1/5, 1)
3. (1/5, 0)
4. (1/5, −1)

Correct answer: (1/5, 0)

Solution

The integral of tanⁿ x can be expressed in terms of lower powers of tan x and x, and through integration by parts or reduction formulas, it can be shown that the combination I₄ + I₆ results in a term proportional to tan⁵ x with a coefficient of 1/5, while the x⁵ term does not appear, leading to the ordered pair (1/5, 0).

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