Let $f(x)$ be the frequency distribution of the variable $x$. Let assume that $\int^{\infty}_{-\infty} f(x) ≠ 1$. Let $g(x) = C f(x)$ such as $C$ is the constant of integration so that $\int^{\infty}_{-\infty} g(x) = 1$.
Are the shapes of $f(x)$ and $g(x)$ exactly the same (only the scale of the density of frequeny changes) or might they be different? Does it mean that a constant of integration necessarily need to be independent of the random variable $x$?
For $f(x)$ to be a frequency distribution, it requires that $\int_{-\infty}^{\infty}f(x)dx = 1$. So your question is moot.