Let $X=Z(f)$ be a curve in the affine plane with a unique singular point $(0,0)$. Let's keep in mind the examples $f=y^2-x^5,\ f=y^2-x^2-x^3 $. Now we want to calculate the normalization of $R=k[x,y]/f.$ The usual method to do this is to guess an integral fraction $t$ and to show that $R[t]$ is integrally closed, so that it must equal the normalization of $R$. $t=\frac{y}{x^2}, \ t=\frac{y}{x}$ respectively for the above mentioned examples.
My question is: In general, how many fractions will we have to add? is this an invariant of a singular point?
I am tempted to say that in this case it is equal to the amount of blow-ups one needs to do to make $X$ regular.