Let f(x) = x - [x] for every real number x , where [x] is integral part of x . then _ -1 ¹ f(x) , dx is

Step-by-step solution

Given f(x) = x - [x] The integral is _ -1 ¹ (x - [x]) , dx This can be separated as _ -1 ¹ x , dx - _ -1 ¹ [x] , dx Since x is an odd function, _ -1 ¹ x , dx = 0 For the second integral, we split the limits at the integer value 0 : _ -1 ¹ [x] , dx = _ -1 ⁰ [x] , dx + ₀¹ [x] , dx In the interval (-1, 0) , [x] = -1 and in the interval (0, 1) , [x] = 0 _ -1 ¹ [x] , dx = _ -1 ⁰ (-1) , dx + ₀¹ 0 , dx = -1[x]_ -1 ⁰ + 0 = -1(0 - (-1)) = -1 Substituting these back, we get: _ -1 ¹ (x - [x]) , dx = 0 - (-1) = 1 Answer: 1