Classical Action for Harmonic Oscillator

DeclanTKatt

1. The problem statement, all variables and given/known data
Hello. I am attempting to evaluate the classical action of a harmonic oscillator by using the Euler-Lagrange equations.

2. Relevant equations
The Lagrangian for such an oscillator is

L=(1/2)m([tex]\dot{x}[/tex]²-[tex]\omega[/tex]²x²)

This is easy enough to solve for. The classical action is defined by S_cl=[tex]\int[/tex]L dt

3. The attempt at a solution
I know what the answer is, but I am having difficulty achieving it. So far I have used:
x=sin([tex]\omega[/tex]t)
[tex]\dot{x}[/tex]=[tex]\omega[/tex]cos([tex]\omega[/tex]t)

Substituted these into the Lagrangian and then integrated, with respect to t, for the classical action. This did not provide the proper results.

Any suggestions would be greatly appreciated. Thanks