If (x-1) d x (x+1) x^3+x^2+x =A ⁻¹ f(x) + constant, then the ordered pair (A, f(-1))=

Step-by-step solution

We have, (x-1) d x (x+1) x^3+x^2+x =A ⁻¹ f(x) +C Let aligned I & = (x-1) d x (x+1) x^3+x^2+x & = x-1 x(x+1) x+ 1 x +1 d x aligned Put, aligned x+ 1 x +1 & =t and (1- 1 x^2 ) d x=d t & = (x-1) x(x+1) t x^2 (x^2-1 ) d t & = x (x+1)^2 t d t= d t x^2+2 x+1 t & = d t (1+x+ 1 x +1 ) t = d t (1+t) t = & 2 ⁻¹ t +C=2 ⁻¹ x+ 1 x +1 +C aligned aligned & A=2 f(x)=x+ 1 x +1 & A=2 f(-1)=-1-1+1=-1 & (A, f(-1)=(2,-1) & aligned