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Evaluate the definite integral from x = 0 to x = 2 of the greatest integer function applied to (x² - x + 1), i.e., compute the integral of [x² - x + 1] dx over [0, 2], where [t] denotes the floor (greatest integer not exceeding t).

1. 2 + sqrt(5)
2. sqrt(5)/3 + 2
3. (5 + sqrt(5))/2
4. (5 - sqrt(5))/2

Correct answer: (5 - sqrt(5))/2

Solution

On (0,1) the floor is 0, on (1, (1+sqrt(5))/2) the floor is 1, and on ((1+sqrt(5))/2, 2) the floor is 2; summing the sub-integrals gives 0 + ((sqrt(5)-1)/2) + 2*(2 - (1+sqrt(5))/2) = (5 - sqrt(5))/2.

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