Since ##v(t) = \int -\cos t \hat i -\sin t \hat j dt, ## you get ##v(t) = -\sin t \hat i + \cos t \hat j + \vec C ##.
Note that C can be any constant vector.
Plug in t = 0 and compare with your v(0) term to solve for vector C.
##v(0) = -\sin 0 \hat i + \cos 0 \hat j + \vec C = 0 \hat i + 1 \hat j + \vec C = \hat j + \hat k ##
This gives you ##\vec C = \hat k ##. Put this back into your equation for v(t) and you get ##v(t) = -\sin t \hat i + \cos t \hat j + \hat k ##.
Next, you integrate velocity to get position.
##r(t) = \int -\sin t \hat i + \cos t \hat j + \hat k dt = \cos t \hat i + \sin t \hat j + t \hat k + \vec K. ##
Where, again, vector K is any constant vector.
As before, put in t = 0 and compare with initial position ## \hat i ## to solve for the constant vector K.
##r(0) = \hat i = \cos 0 \hat i + \sin 0 \hat j + 0 \hat k + \vec K =1 \hat i + 0 \hat j + 0 \hat k + \vec K . ##
This shows you that vector K is the zero vector, so you can write r(t) as
##r(t) = \cos t \hat i + \sin t \hat j + t \hat k . ##