Our first step in this question is to re-write the exponents so it is easier to integrate:

`int (3x^2 + sqrt[x] - frac{5}{x^3})dx`

`=int (3x^2+x^[frac{1}{2}]-5x^-3)dx`

`=x^3+frac{x^[frac{3}{2}]}{3/2}-5frac{x^-2}{-2}+K`

`=x^3+frac{2x^[frac{3}{2}]}{3}+frac{5x^-2}{2}+K`