Consider the integrals A= ₀^ x d x x+ x and B= ₀^ x d x x- x What is the value of B ?

Step-by-step solution

aligned & B= ₀^ x x- x ( x+ x) ( x+ x) d x & =- ₀^ ^2 x+ x x 2 x d x & = -1 2 ₀^ 2 ^2 x 2 x d x- 1 2 ₀^ 2 x x 2 x d x & = -1 2 ₀^ 1- 2 x 2 x d x- 1 2 ₀^ 2 x d x & = -1 2 ₀^ 2 x d x+ 1 2 ₀^ d x- 1 2 ₀^ 2 x d x & = -1 2 [ | 2 x+ 2 x| 2 ]₀^ + 1 2 [x]₀^ - 1 2 [ | 2 x| 2 ]₀^ & = & -1 4 [ (1+0)- (1+0)]+ 1 2 [ +0]- 1 4 [ (1)- (1)] aligned aligned & = & -1 4 [ (1+0)- (1+0)]+ 1 2 [ +0]- 1 4 [ (1)- (1)] & =0+ / 2-0= / 2 aligned