Getting the original function from a derivative

Question

i know that $f_x = 4x^3 + 4xy $ is the partial derivative of the function $ f = x^4 + 2x^2y $

But how is the original function $ f = x^4 + 2x^2y $ found from the partial derivative ?

As in, how do i reverse the process?

Answer 1

Rule: We see $y$ as a constant. Hence we have

$$f(x)=\int f_x dx=x^4+2x^2y+g(y)$$ where $g(y)$ is any function on $y$.