Should I still be with him?

My boyfriend and I have been dating for 8 months today. And back in march I found out he cheated on me with this girl he knew. We broke up for a week and then got back together. Well last week I saw these texts between him and a girl that were not appropriate. He said he was drunk when he texted her. We've talked about it. I had him delete her number and any other girls he hasnt spoken to in a while But ever since I get sketchy about who he's texting and what's he doing when I'm not with him. I half way trust him. I already know what most answers would be but I'd just like more points of view. I love this kid to death though. Don't ask why.