₀¹ ⁻¹ [ 2 x-1 1+x-x² ] d x=

Step-by-step solution

Let aligned I &= ₀¹ ⁻¹ [ 2 x-1 1+x-x² ] d x &= ₀¹ ⁻¹ [ 2 x-1 1+x(1-x) ] d x= ₀¹ ⁻¹ [ x-(1-x) 1+x(1-x) ] d x &= ₀¹ [ ⁻¹ x- ⁻¹(1-x) ] d x ...(1) &= ₀¹ ⁻¹(1-x)- ⁻¹[1-(1-x)] d x &= ₀¹ [ ⁻¹(1-x)- ⁻¹ x ] d x ...(2) Equation (1)+(2) gives 2 I=0 I=0 aligned