Why Is the Antiderivative of Cos(x) Different from Its Integral?

[A.J.710]

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

 

[LCKurtz]

I am just stuck on the fact that the antiderivative of cos(x) is -sin(x)

No. That is the derivative on cos(x), not its antiderivative.

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

 

[A.J.710]

OMG no wonder. I can't believe I have been tripping over something that simple. I'm just so overwhelmed right now I am making stupid mistakes. Thank you for pointing that out.

No. That is the derivative on cos(x), not its antiderivative.