₁^2 x^4-1 x^6-1 d x=

Step-by-step solution

Let I= ₁^2 x^4-1 x^6-1 d x aligned & = ₁^2 (x-1)(x+1) (x^2+1 ) (x-1) (x^2+x+1 )(x+1) (x^2-x+1 ) d x & = ₁^2 x^2+1 (x^2+x+1 ) (x^2-x+1 ) d x aligned = ₁^2 ( 1 2 x^2+x+1 + 1 2 x^2-x+1 ) d x= 1 2 ₁^2 ( 1 (x+ 1 2 )^2+ ( 3 2 )^2 + 1 (x- 1 2 )^2+ ( 3 2 )^2 ) d x= 1 2 2 3 ⁻¹ ( 2 x+1 3 )+ ⁻¹ ( 2 x-1 3 ) ₁^2= 1 3 ⁻¹ ( 5 3 )+ ⁻¹( 3 )- ⁻¹( 3 ) . .- ⁻¹ ( 1 3 ) = 1 3 ⁻¹ ( 5 3 - 1 3 1+ 5 3 ) = 1 3 ⁻¹ ( 3 2 )