Verified answer

If F = ∇f, then we have by equating like entries:

f_x = 2xy^3+ 3yz + z cos(xz) + 2x,

f_y = 3x^2 y^2 + 3xz + z sin(yz) + 2y,

f_z = 3xy + x cos(xz) + y sin(yz) + 2z.

Integrate these with respect to the appropriate variables, we obtain (respectively)

f = x^2 y^3 + 3xyz + sin(xz) + x^2 + g(y,z)

f = x^2 y^3 + 3xyz - cos(yz) + y^2 + h(x,z)

f = 3xyz + sin(xz) - cos(yz) + z^2 + k(x,y).

Putting this all together (compare the three versions of f), we obtain

f(x,y,z) = x^2 y^3 + 3xyz + sin(xz) - cos(yz) + x^2 + y^2 + z^2 + C.

I hope this helps!