Implicit differentiation? implicit integration?

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Main Question or Discussion Point

We have implicit differentiation:
i.e. x^2 + y^2 = 7
-> 2x +2y(dy/dx) = 0.
and solve for dy/dx gives you the derivative of y with respect to x

However, is there not implicit integration?

for terms linear in y,
i.e. x^2 + y = 7
-> X^3/3 +int(y) = 7x + K,
and solve for int(y) to get the intergral of y with respect to x

But what about terms non-linear in y?
 

Answers and Replies

HallsofIvy
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Implicit integration is basically the use of the "chain rule": if you have y2 then d(y2)/dx= d(y2)/dy dy/dx= 2y y'. Unfortunately, you cannot, in general go the opposite way: using the chain rule to differentiate you calculate the expression, dy/dx, that must be multiplied, while integrating it has to already be in the integral. "Implicit" integration is basically "substitution" which only works if the derivative of the function substituted is already in the integral.
 

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