If I am in my first trimester of pregnancy, can I still do my regular exerise and run 5 miles daily?

Yes. Particularly early in pregnancy, exercise is great for staying in shape and healthy for when the baby comes. As you get closer to term, you will want to move to lower impact aerobics, such as walking and swimming, but as long as you feel good, keep it up!
Usually you can. Please first check with your own physician and ob/gyn before doing so.