If you integrate as an indefinite integral it will be -cos(x) + C.
If you want the area (definite integral) between zero and x then it will be -cos(x) - -cos(0) = -cos(x) + 1.
If you said, "the area function must take the form -cos(x) + C, and I want to measure the area from zero", then the area function must start at zero when x=0, and that leads to the same conclusion, ie that C = 1, and the area function is -cos(x) + 1.
If you wanted to measure the area between say pi/2 and x, then the answer would be different.
@justrichmal I'm super into trig atm because of my line of work - if you've not already seen it, check out how a rotating vector makes a sine wave and how the triangles and angles come into it. So cool!