In Reid's Undergraduate Commutative Algebra he shows that the normalization of $A = k[X,Y]/(Y^2 - X^3)$ is $k[t]$, where $t = y/x$, because the field of fractions of $A$ is $k(t)$, $t \in k(t)$ is integral over $A$ because $x = t^2$, but is not in $A$ itself, and $k[t]$ is a UFD and therefore normal.
In the chapter exercises, Reid asks for the calculation of the normalizations of $k[X,Y] / (Y^3 - X^5)$ and $k[X,Y] / (Y^5 - X^{19})$. I would guess they must be different than the earlier example, but I don't see how: isn't the field of fractions over each also $k(t)$, where $t = y/x$? And then $t^3 = x^2$ in the first and $t^5 = x^{14}$ in the second, so $t$ is integral over both rings, and again $k[t]$ is a UFD and therefore normal. Do both of these have the same normalization as the first example? If they do, am I right in saying that that's because both the curves $y^5 = x^3$ and $y^5 = x^{19}$ can also be parameterized into a curve over $t$ with no singularities?