let say we have $z(x,y)$, and $x(t), y(t)$
From total differential we have $$dz=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy$$
Then we differentiate it with respect to $t$:
$$\frac{dz}{dt}=\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}$$
Then I integral both sides with respect to $t$ again
$$\int\frac{dz}{dt}dt=\int\frac{\partial z}{\partial x}\frac{dx}{dt}dt+\int\frac{\partial z}{\partial y}\frac{dy}{dt}dt$$
$$\int dz=\int\frac{\partial z}{\partial x}dx+\int\frac{\partial z}{\partial y}dy=z$$
From the last line, I notice this is incorrect because we cannot integrate w.r.t different variables at the same time, but from the last equation, I integrate with $t$ on both sides, and this should be fine, so I don't see where it went wrong.
From the second equation, $\frac{dz}{dt}=\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}$, how do I recover $z$ correctly?
I read this, the last post but I don't see how to from $df=xdy+ydx$ to $\int\frac{\partial f}{\partial y}dy=\int xdy+\int y\frac{\partial x}{\partial y}dy$ How to integrate a total derivative?