# Implicit derivative, implicit integral?

An undergraduate student of mine asked me last year the following question.

Let $f:\mathbb R^2\to \mathbb R$. The equation $f(x,y)=0$ defines implicitly a function $y:\mathbb R\to\mathbb R$ and we can express its derivative in terms of the partial derivatives of $f$. Now, is there any method to express an "implicit integral" of $y$ in terms of other quantities? I.e. $$\int_{ I} y(x) dx$$ for some interval $I$ or as an antiderivative as a function of $y$, in terms of integrals of $f$ in some sets of $\mathbb R^2$.

I found it was an interesting question, and I congratulated her because of her out of the box thinking. But I did not know how to answer this, my intuition is that no such method exists.

I'm not sure to well interpret your question, but if we search a function $F(x)$ such that $F'(x)=y(x)$ than the implicit function $f(x,y)=0$ becomes a differential equation: $f(x,F')=0$ that can be solved (when it is possible) with one of the classical methods.