Relation between Potential Field, Force, Kinetic Energy and Abs.Energy

[EEristavi]

Homework Statement

A single conservative force acts on a 5.00-kg particle
within a system due to its interaction with the rest of the
system. The equation Fx = 2x + 4 describes the force,
where Fx is in newtons and x is in meters. As the particle
moves along the x axis from x = 1.00 m to x = 5.00 m,
calculate (a) the work done by this force on the particle,
(b) the change in the potential energy of the system, and
(c) the kinetic energy the particle has at x 5 5.00 m if its
speed is 3.00 m/s at x 5 1.00 m.

Relevant Equations

W = ∫ F dx
F = -∇U

I understand that the work done is Change of Energy.
W = ∫_xi^x_f F dx = ΔEThe force is gradient of potential energy
F = -∇U (For conservative forces of course)

from here, we can say that change of potential energy is W:
ΔU = -W
but also
ΔU = -W = ΔE

I'm little bit lost here..
Can you help me understand this topic? where I make mistake?

 

[Orodruin]

from here, we can say that change of potential energy is W:
ΔU = -W

Don't confuse the work done on the object by the conservative force with the work you need to do against the conservative force to lift the object.

 

[Abhishekoo7]

You r on right way