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Question: Let f (x) = x- \([ \mathrm { x } ]\) , for every real number x, where \([ \mathrm { x } ]\) is in...

Let f (x) = x- [x][ \mathrm { x } ] , for every real number x, where [x][ \mathrm { x } ] is integral part of x. Then ∫−11f(x)\int _ { - 1 } ^ { 1 } \mathrm { f } ( \mathrm { x } ) dx is.

A

1

B

2

C

0

D

½

Answer

1

Explanation

Solution

∫−11(x−[x])dx=∫−10(x−[x])dx+∫01(x−[x])dx\int _ { - 1 } ^ { 1 } ( x - [ x ] ) d x = \int _ { - 1 } ^ { 0 } ( x - [ x ] ) d x + \int _ { 0 } ^ { 1 } ( x - [ x ] ) d x

=

=