@Luna1980,
I am so sorry you are in this place.
Right now, your husband is not showing you he is truly committed to reconciling. You need to set down some boundaries if you are to stay.
He needs to send a no contact letter or text to the other woman. It should include a statement about how he loves you, is committed to working on the marriage, wants nothing to do with her, and how what he did was a mistake.
He also needs to completely open up all electronics with you - you need full and complete access to all his email and apps. This is necessary to build trust.
He also needs to agree not to go on any further business trips. If they work in the same place, he needs to start looking for a new job.
If he will not do these things, then no, you should not stay, and no, you cannot trust him.
I understand how difficult this is. I have been there, and it's world shattering. Please take care of yourself. Try to eat and drink, and be kind to yourself. He did not cheat because of you. He cheated because he was selfish.