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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.

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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.

Clearly there is not a simple method to integrate the ODE (based on given rules as for the implicit differentiation). but this is not different from the usual difference between differentiation ad integration of an ''explicit'' function.

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  • $\begingroup$ Of course! I'll come back to my student with this, thx $\endgroup$ – Tal-Botvinnik Apr 13 '18 at 16:35

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