If 1-x^2 x^4 ~d x= A (x) ( 1-x^2 )^ m + c for a suitable chosen integer m and a function A (x) , where c is a constant of integration, then ( A (x))^ m equals

Step-by-step solution

Let aligned I & = 1-x^2 x^4 ~d x & = x 1 x^2 -1 x^4 ~d x & = 1 x^2 -1 x^3 ~d x aligned Let 1 x^2 -1= t aligned -2 x^3 & ~d x= dt 1 x^3 ~d x= - dt 2 I & =- 1 2 t dt & = -1 2 ( t )^ 3 2 3 2 + c & = -1 3 ( 1 x^2 -1 )^ 3 2 + c & = -1 3 (1-x^2 )^ 3 2 (x^2 )^ 3 2 + c & = -1 3 ( 1-x^2 )^3 x^3 aligned Comparing with A (x) ( 1-x^2 )^ m + c , we get gathered A (x)= -1 3 x^3 and m =3 ( A (x))^ m = ( -1 3 x^3 )^3= -1 27 x^9 gathered