Energy in a hanging rope

Consider a rope of mass m, natural length L and modulus E where k = E/L.
Suppose a point P is at a distance x from the top before extension.
Length of rope below P when unstretched = L-x
Weight of rope below P when unstretched = [L- x]/Lmg

And this will accordingly be the tension at P when the rope is suspended.

Now consider an element of length δx at P, and suppose the extension of this element is e.

Then we have: ([L-x]/L)mg = Ee/δx x

Therefore energy of the element = Ee²/2δx x = ([(L- x)²m²g²]/2EL²) δx x

The total energy in the hanging rope is the integral of this from 0 to L.