Evaluate the following integrals : `int(x^(-1/3)+sqrt(x)+2)/(root(3)(x))`

Explanation

`int(x^(-1/3)+sqrt(x)+2)/(root(3)(x))`

`int frac{x^(-1/3)+sqrt x+2}{x^(1/3)} dx`

`=int x^(-1/3-1/3)+int x^(1/2-1/3)+2 int x^(-1/3) dx`

`=int x^(-2/3)+int x^(1/6)+2 int x^(-1/3)dx`

We know that `int x^n dx=frac{x^(n+1)}{n+1}+C`

`=frac{x^(-2/3+1)}{-2/3+1}+frac{x^(1/6+1)}{1/6+1}+2 frac{x^(-1/3+1)}{-1/3+1}+C`

`=3x^(1/3)+6/7x^(7/6)+3x^(2/3)+C`