I know this has been asked before but the answers didn't make sense to me. Can someone explain, preferably with a rigorous proof, why the single integral of a partial derivative of a function equals that function plus a function of the other variable? I want a general answer so one variable may be dependent on the other. ex: f[x(t), t]
Integrating a Partial Derivative which doesn't actually answer the question. It only gives the answer for specific cases.
https://math.stackexchange.com/a/606708/333261 gives the answer but it's explanation makes no sense to me. Why does it matter that y^2 goes to 0 after the partial? Additionally, I believe it assumes x and y to be independent variables so it is not a general answer.
Integrating a partial derivative with second variable a function of the first?, Does an integral of a partial derivative make the partial derivative disappear?, just state the answer. Their explanations don't make sense. I don't even understand their explanations enough to state what I don't understand about them.
