For any real number x , let [x] denotes the largest integer less than or equal to x . Let f be a real valued function defined on the interval [-10,10] by f(x)= array cc x-[x] & if [x] is odd 1+[x]-x & if [x] is even array . . Then the value of ^2 10 _ -10 ¹⁰ f(x) x d x is

Step-by-step solution

Given, f(x)= array cc x-[x] & if [x] is odd 1+[x]-x & if [x] is even array .f(x) and x both are periodic with period 2 and both are even. aligned & _ -10 ¹⁰ f(x) x d x & =2 ₀¹⁰ f(x) x d x aligned aligned & =10 ₀^2 f(x) x d x & Now, ₀^1 f(x) x d x & = ₀^1(1-x) x d x=- ₀^1 u u d u & and ₁^2 f(x) x d x & = ₁^2(x-1) x d x=- ₀^1 u u d u & _ -10 ¹⁰ f(x) x d x & =-20 ₀^1 u u d u= 40 ^2 & ^2 10 _ -10 ¹⁰ f(x) x d x=4 aligned