₀^ 2 | x- x| d x has the value

Step-by-step solution

aligned I & = ₀^ 2 | x- x| d x & = ₀^ 4 -( x- x) d x+ _ 4 ^ 2 ( x- x) d x aligned aligned f (x) & = f (-x) & & 0 x 4 & = f (x) & & 4 x 2 aligned aligned & = ₀^ 4 ( x- x) d x+ _ 4 ^ 2 ( x- x) d x & =( x+ x)₀^ 4 +(- x- x)_ 4 ^ 2 & = [ 4 + 4 - 0- 0 ] aligned - [ 2 + 2 - 4 - 4 ] aligned & = [ 1 2 + 1 2 -0-1 ]- [0+1- 1 2 - 1 2 ] & = [ 2 2 -1 ]- [1- 2 2 ] & = 4 2 -2 & = 4 2 2 -2 & =2 2 -2 & =2( 2 -1) aligned