Spring mass system with applied force kinetic energy

[fobos3]

I'm trying to work out a model for a spring mass system with a force acting at the centre of mass of the mass using Lagrangian mechanics. I can't work out the kinetic energy. I know the kinetic energy \text{KE}=\dfrac{1}{2}mv^2. I also have W=\int_a^b F \, dt.
Should I use \Delta \text{KE} + \Delta \text{PE} =W

Some help will be appreciated.

P.S. I'm using Lagrangian mechanics because later I'm planning some calculations on double spring systems with applied force.

 

[Triangulum]

W = \int \vec{F} \cdot \vec{dx}

The expression you had was for Impulse not Work