Antiderivative vs integral

I am just stuck on the fact that the antiderivative of cos(x) is -sin(x) but the integral of cos(x) is a positive vale of sin(x). Can someone please explain this to me. I am two weeks behind on my calculus course now because I got stuck on integration by parts and don't understand anything after that and I am reading over my teacher's answer key and see this in one of the solutions and don't understand why. Thanks