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Evaluate the integral I = ∫₀¹ x(1−x)ⁿ dx.

1. 1/(n+1) + 1/(n+2)
2. 1/(n+1)
3. 1/(n+2)
4. 1/(n+1) − 1/(n+2)

Correct answer: 1/(n+1) − 1/(n+2)

Solution

The correct option is derived from the integration of the function x(1−x)ⁿ, which can be solved using integration by parts or the beta function. The result simplifies to 1/(n+1) - 1/(n+2), reflecting the contributions from both the polynomial and the limits of integration.

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