If [x] is the greatest integer x , then the value of the integral _ -0.9 ^ 0.9 ( [x^2 ]+ ( 2-x 2+x ) ) d x is

Step-by-step solution

aligned & _ -0.9 ^ 0.9 [x^2 ]+ ( 2-x 2+x ) d x = & _ -0.9 ^ 0.9 [x^2 ] d x+ _ -0.9 ^ 0.9 ( 2-x 2+x ) d x aligned aligned & =0+ _ -0.9 ^ 0.9 ( 2-x 2+x ) d x & Put x=-x f(x)= 2-x 2+x & and f(-x)= 2+x 2-x & =- (2-x) 2+x =-f(x) aligned So, it is an odd function, hence Required integral =0 .