In the above image, there is the Special Field Research page open, notice on the side there are 3 different tasks, which you can earn 5000 XP (each) by completing. However, when I've completed tasks such as these, I can't tell if I actually received the 5000 XP. Any help would be greatly appreciated.
In general, when you gain experience in Pokemon Go and then go back to the main map screen a number will appear in the bottom left above your character picture that indicates how much experience you have gained. Unfortunately, XP rewards from special research are the exception to this; they will display no information there after collecting them. The only way you can verify that you are getting the XP is by checking your total experience value before and after collecting it. You can see that number by tapping your trainer picture top open your trainer information page and then scrolling all the way to the bottom.
Note that in general, once a research task is complete, it turns orange, then you can tap it to collect the reward. So if you do one of those tasks, you won't actually get the experience until you collect it.