Your eyes do integrate actually - it's why a CRT TV works. A CRT is not actually illuminated all at once - it illuminates one line at a time, but the scan frequency is fast enough that your eyes basically integrate it into a single, continuous image. However, the period of time over which your eyes integrate is fixed, and it's fairly short. Since you're seeing in real time, it wouldn't make sense for your eyes to integrate over periods of minutes for example, since you will almost never stare at one spot for multiple minutes at a time. However, photographic plates can be used where they are observing a single object for long periods of time, which allows the integration to be used over much longer periods than the eye does. This allows for greatly increased sensitivity by comparison.
Also, the eye is pretty darn sensitive. As photon count decreases, the eye responds not by integrating over longer periods the way a camera would (which would decrease ability to see motion), but instead by increasing the sensitivity of the retina and opening up the aperture to let in more photons. A fully dark adapted eye can see a remarkably dim source, so for nearly any situation, long integration isn't really necessary, and as stated above, it would actually hurt the ability to perceive motion.