Verified answer

Checking whether or not dP/dy = dQ/dx only verifies if it is conservative or not.

f(x,y) = x^2ln(y) + ycos(x) + K (constant)

I remember back when I took the class there was a methodical way of solving this, but I never used the method my textbook suggested. I find it easiest to intuitively visualize what f(x,y) is if ∇f(x,y) = F(x,y). Just integrate the "i" component of the F vector with respect to x, and then integrate the "j" component with respect to y, and you should get your f(x,y) function.