The value of _ 1 3 ^1 (x-x^3 )^ 1 3 ~d x is

Step-by-step solution

Evaluate ( _ 1 / 3 ^1 (x-x^3 )^ 1 / 3 d x ) Rewrite the integrand: (x-x^3=x (1-x^2 ) ) On ([1 / 3,1] ), the expression is non-negative, so ( (x-x^3 )^ 1 / 3 =x^ 1 / 3 (1-x^2 )^ 1 / 3 . ) Use substitution (x= ^ 2 / 3 ), which gives (x^ 1 / 3 = ^ 2 / 3 , 1-x^2= ^2 , ) and the integral simplifies perfectly to ( d = 1 2 ^2 . ) Converting bounds: ( gathered x= 1 3 ^ 2 / 3 = 1 3 = 1 3 , x=1 =1 . gathered ) Thus value is ( 1 2 [1^2- ( 1 3 )^2 ]= 1 2 (1- 1 3 )= 1 2 2 3 =0 . )