The value of the integral, ∫ 1 3 x 2 - 2 x - 2 d x , where x denotes the greatest integer less than or equal to x , is

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∫ 1 3 x - 1 2 - 3 d x = ∫ 0 2 x 2 d x - 3 ∫ 1 3 d x = ∫ 0 1 0 d x + ∫ 1 2 1 · d x + ∫ 2 3 2 · d x + ∫ 3 2 3 · d x - 6 = 2 - 1 + 2 3 - 2 + 3 2 - 3 - 6 = - 2 - 3 - 1

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