Feynman Path Integral Evaluation of the Propagator

The propagator for harmonic oscillator can be evaluated as follows:

$<x|{\hat {U}}(t,0)|x_{0}>=e^{{\frac {i}{\hbar }}S}\int _{y(0)=0}^{y(t)=0}D[y(t')]e^{{\frac {i}{\hbar }}\int _{0}^{t}({\frac {1}{2}}my'^{2}-{\frac {1}{2}}ky^{2})dt'}$

where $y(t')$ is the deviation of possible trajectories about the classical trajectory.