integral substitution

Question

x(x-6)^(1/3) dx

Answer 1

Let y=x-6 then dy=dx and x=y+6.

Substituting y in place of x: (y+6)y^(1/3)dy=(y^(4/3)+6y^(1/3))dy.

Integrating wrt y we get: 3y^(7/3)/7+6y^(4/3)*3/4+C where C is constant of integration.

Replacing y with x: (3/7)(x-6)^(7/3)+(9/2)(x-6)^(4/3)+C=3(x-6)^(4/3)((x-6)/7+(3/2))+C.

This simplifies to (3/14)(x-6)^(4/3)(2x+9)+C.