The value of ₀^1 ⁻¹ ( 2 x-1 1+x-x^2 ) d x is

Step-by-step solution

aligned & Let I= ₀^1 ⁻¹ ( 2 x-1 1+x-x^2 ) d x & = ₀^1 ⁻¹ [ x+(x-1) 1+x(1-x) ] d x= ₀^1 ⁻¹ [ x+(x-1) 1-(x-1)(x) ] d x & = ₀^1 [ ⁻¹ x+ ⁻¹(x-1) d x= ₀^1 ⁻¹ x d x+ ₀^1 ⁻¹(x-1) d x . & .= [ [x ⁻¹ x ]₀^1- 1 2 ₀^1 2 x 1+x^2 d x ]+ [x ⁻¹(x-1) ]₀^1- 1 2 ₀^1 2 x-2+2 1+(x-1)^2 d x ] & = ( 4 )- 1 2 [ |1+x^2 | ]₀^1+0- 1 2 ₀^1 2 x-2 1+(x-1)^2 d x- ₀^1 d x 1+(x-1)^2 & = ( 4 )- 1 2 2- 1 2 [ |1+(x-1)^2 | ]₀^1- [ ⁻¹(x-1) ]₀^1 & = 4 - 1 2 2- 1 2 (0- 2)- [0- ⁻¹(-1) ] & = 4 - 1 2 2+ 1 2 2- 4 =0 aligned