Simplify and find the general antiderivative of a rational function

Split the fraction term by term: $h(x) = \dfrac{x^3}{x} + \dfrac{2x}{x} = x^2 + 2$, valid for $x \neq 0$. This removes the fraction and leaves a simple polynomial to integrate.

Apply the power rule to each term: $\int x^2\,dx = \dfrac{x^3}{3}$ and $\int 2\,dx = 2x$. Adding the arbitrary constant of integration gives the general antiderivative $H(x) = \dfrac{x^3}{3} + 2x + C$.

Because differentiation kills constants, any constant added to the simplified function x squared plus two vanishes when you take a derivative. So when reversing the process to find the antiderivative, infinitely many constants are equally valid. Writing plus C accounts for every possible constant that could have been present originally. This is why the plus C is required, not optional.