I am having a Nikon D5200, when the flash pops out the shutter burst mode can't able to apply. Is it possible to take photos in burst mode by using flash In Nikon D5200?
With some cameras it can theoretically be yes. With most Nikon DSLR cameras the flash is limited to one discharge per shutter button press.
With pretty much all DSLRs with built in flashes it is, practically speaking, no.
The reason is that the flash has to be charged before it fires. If the flash is totally discharged as it would be with a full power flash dump, it might take a couple of seconds to re-energize the capacitor that stores the charge for the flash.
If you're using automatic flash, though, the some cameras may decide you don't need full power. The less power the flash uses for each shot, the more energy is left in the capacitor and the less time it takes to recharge the capacitor back to full capacity. So if you're shooting under conditions where not much flash power is needed the flash may be able to keep up with the camera for a few frames.
With built-in flashes the capacitor usually must be fully charged before the flash will fire again. With some external flashes, the flash might go ahead and fire even when partially charged. Of course this means the total flash power available will be less than if the capacitor is full. Some cameras will allow the shutter to fire again without the flash going off until it is charged again. You'll get pictures at the camera's burst rate, but you won't get the benefit of the flash's light for most of them.
To get a flash system that can keep up with a camera shooting at 5 frames-per-second you'll need a lot of power available to recharge the flash's capacitor. This usually requires an external battery pack that is larger and bulkier than your camera. It may even require a wall power outlet to charge the capacitor. Even then, such a rate can not be maintained for long. This is due to the amount of heat that builds up. Once the flash heats up it must be allowed to cool before being used again.
Not with the internal flash.
In continuous shutter mode, the internal flash will fire only once each time that you press the shutter button. Most of the Nikon manuals specify those exact words, but I am surprised that I don't find it in the D5200 manual. For example, page 148 of the D7200 manual, or page 186 of D800 manual.
Speedlight flashes must recycle to fire again correctly, however, most hot shoe flashes can recycle that fast if at low power level. But hot shoe flashes will try, regardless if they are able to recycle or not (has to be low power to work).
But the internal flash is different, handled more carefully.