The electrons injected into a laser wakefield undergo betatron oscillation and give rise to the emission of intense X-ray radiation. To investigate the generation conditions of the X-rays, the relativistic motion of an electron injected in an off-axis position has been simulated with wakefield profiles which are pre-calculated with a two-dimensional particle-in-cell code. The wakefield with a plasma density of $1.78{ imes}10^{18};cm^{-3}$ is generated by the laser with an intensity of $1.37{ imes}10^{18};W/cm^2$ and a pulse width of 30 fs. From the calculation of the single particle motion, the characteristics of the betatron radiation are investigated in the time domain. As the transverse injection position increases, the power and the duration time of the radiation increase, but the width of each pulse decreases.