I have often seen integrals such as
$$A = \int_{t=0}^{t=T} \frac{\partial}{\partial t} \phi(t,x) dt$$ and I'm wondering if the integral cancels the partial derivative when the variable that the $\frac{\partial}{\partial t}$ is the same as the $dt$ variable?
Does the integral cancel with the partial derivative or do both remain? And, does the answer to that question change based if the $dt$ term had been $dx$?
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