If (x^2-1 ) (x+1)^2 x (x^2+x+1 ) d x=A ⁻¹ ( x^2+x+1 x )+C , in which C is a constant, then A equals to

Step-by-step solution

aligned & Let I= x^2-1 (x+1)^2 x (x^2+x+1 ) d x & Now, d d x ( ⁻¹ x^2+x+1 x ) & = 1 1+ x^2+x+1 x d d x ( x^2+x+1 x ) & = x x^2+2 x+1 [ x 2 x+1 2 x^2+x+1 - x^2+x+1 1 2 x ( x )^2 ] & = x x^2+2 x+1 [ x (2 x+1) 2 (x^2+x+1 ) - x^2+x+1 2 x ] & = x (x^2+2 x+1 ) [ (2 x+1) x- (x^2+x+1 ) 2 x (x^2+x+1 ) x ] & = x (x+1)^2 [ x^2-1 2 x x (x^2+x+1 ) ] & = x^2-1 2(x+1)^2 x (x^2+x+1 ) & aligned aligned I & = d d x (2 ⁻¹ x^2+x+1 x ) & =2 ⁻¹ ( x^2+x+1 x )+C A & =2 aligned