The differential equation $\left(4xy+1\right)dx+\left(2x^2+\cos\left(y\right)\right)dy=0$ is exact, since it is written in the standard form $M(x,y)dx+N(x,y)dy=0$, where $M(x,y)$ and $N(x,y)$ are the partial derivatives of a two-variable function $f(x,y)$ and they satisfy the test for exactness: $\displaystyle\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$. In other words, their second partial derivatives are equal. The general solution of the differential equation is of the form $f(x,y)=C$

We have found our $f(x,y)$ and it equals

Then, the solution to the differential equation is

Group the terms of the equation