How do you integrate the second derivative of a function?

I know when you integrate the first derivative of a function you just get the function, because integrals and derivatives are inverses of each other, (so the integral of dx/dt is x) but when you integrate the second derivative you don't get the first derivative.

According to wolfram alpha the integral of x'' is x*x'''

Why is this? why is the integral of x'' not just x' ?

The reason I ask this is because if you assume the integral of x'' is x' + c

then the derivation of kinetic energy doesn't work.

F = Force

M = Mass

x = Distance

Force = mass * acceleration and

Energy = the integral of force with respect to distance:

F = M * x''

E = integral ( f .dx )

E = integral ( M * x'' .dx )

E = M * integral ( x'' .dx )

E = M * x'

Which isn't true. E should equal 0.5*M*(x')^2

Why does this not work?

I can see that if you do everything in terms of velocity instead of distance this derivation works. But it should work if you do it in terms of distance shouldn't it?