Welcome to Nikonians! What Release mode are you using? Continuous? Setting the camera in Single frame release mode should solve your problem. See page 35 in the www.nikonusa.com/pdf/manuals/noprint/D5100_ENnoprint.pdf|D5100 Users Manual> Good Luck and Enjoy your Nikons!
What exposure mode are you using? Auto, P, S, A, M, or one of the Vari-Program (Pictogram) modes?
Are you shooting indoors in low light?
In low light, in P, S, A, and some of the Vari-Program modes the camera will choose a slow shutter speed. If you deploy the built in Speedlight the shutter speed should increase. You could also increase the ISO.
In Auto exposure mode, the built in Speedlight should deploy whenever the light is low enough that the exposure will be slow enough to capture a blurred image due to camera shake. Good Luck and Enjoy your Nikons!