$\displaystyle \int (x^{1/3}-3x^{-2/3}+6) dx$
=$\displaystyle x^{4/3}-3(x^{1/3})+6x+C$
is this correct?
What in the world are you going on about in this series of posting?
It appears that you have no idea as to what anti-differentiation is all about.
DO YOU? I doubt that you understand any of this.
If $\displaystyle n\ne -1$ then $\displaystyle \int {x^n = \frac{1}{{n + 1}}x^{n + 1} }$