Question about nonelementary integrals

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Nonelementary integrals are not implicit functions. The discussion clarifies that when performing implicit differentiation, the resulting expression for the derivative includes both the independent and dependent variables, which can sometimes be manipulated into an explicit form. However, integrating an explicit function does not guarantee that the integral can be expressed in terms of elementary functions. The concepts of implicit differentiation and nonelementary integrals are distinct and should not be conflated.

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are nonelementary integrals implicit functions?

ie, when we do implicit differentation, we get an explicit function. What if i go the opposite way, and integrate an explicit function to get an implicit antiderivative?
 
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You seem to be mixing up two unrelated concepts.

When you get the derivative of an implicit function, at first the expression for the derivative will include both the independent variable and the dependent variable. You may be able to manipulate it to get an explicit expression for the derivative.

When you integrate an explicit function the integral may or may be expressible in terms of elementary functions. Implicit has nothing to do with it.
 

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