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Given f x = x | x | for x ≤ - 1 x + 1 + 1 - x for - 1 < x < 1 - x | x | for x ≥ 1 , where [.] denotes the greatest integer function. If I = ∫ - 2 2 f x d x , then 3 I =

Correct answer

8

Step-by-step solution

f x = - x 2 &#160; for &#160; x &#8804; - 1 , &#160; f x = 1 &#160; for &#160; - 1 &#60; x &#60; 0 , f x = 2 &#160; for &#160; x = 0 , f x = 1 &#160; for &#160; 0 &#60; x &#60; 1 &#160; and f x = - x 2 &#160; for &#160; x &#8805; 1 &#8658; f x is even &#8756; I = 2 &#8747; 0 2 f x &#160; d x = 2 &#160; &#8747; 0 1 f x &#160; d x + 2 &#8747; 1 2 f x &#160; d x = 2 &#8747; 0 1 1 &#160; d x + 2 &#8747; 1 2 - x 2 d x = - 8 3 &#8756; &#160; | 3 I | &#160; = &#160; | - 8 | &#160; = 8

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