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For arbitrary $\theta, \psi$, is the following statement true:

$$\iint \frac{\partial}{\partial y}(\theta^2) \frac{\partial \psi}{\partial x} -\frac{\partial}{\partial x}(\theta^2) \frac{\partial \psi}{\partial y}\,dy\,dx = 0$$

And, more generally, given that one can say (neglecting constant of integration) $$\int \frac{d}{dx}f(x)\, dx=f(x)$$ is it true that:

$$\int \frac{\partial}{\partial x}f(x)\, dx=f(x)$$

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