Find $f$ if $\nabla f=8 x y \vec{i}+\left(4 x^{2}+25 y^{4}\right) \vec{j}$ \[ f= \]

Step 1 ：The given function is a gradient vector field. To find the original function $f$, we need to integrate each component of the gradient vector field. The $i$ component is $8xy$ and the $j$ component is $4x^2 + 25y^4$. We integrate each component with respect to its corresponding variable, $x$ for the $i$ component and $y$ for the $j$ component.

Step 2 ：Integrating the $i$ component $8xy$ with respect to $x$ gives $4x^2y$.

Step 3 ：Integrating the $j$ component $4x^2 + 25y^4$ with respect to $y$ gives $4x^2y + 5y^5$.

Step 4 ：Adding these two results together gives the original function $f = 4x^2y + 4x^2y + 5y^5 = 8x^2y + 5y^5$.

Step 5 ：Final Answer: \(f= \boxed{8x^{2}y + 5y^{5}}\)