NUMBER 1.) Well I have no clue if you are supposed to set up that diff. eq like that (I don't know any about them). But those are very easily integrated.
u = 1 + cos(x)
du = -sin(x)
u = e^y + 1
du = e^y
NUMBER 2.) It seems odd... but appears that it should be correct? The integral of the acceleration function is the velocity function, which is what you did. Integrate that again to get the position function. But it does look odd, especially as the variable inside the function is a v. Shouldn't it be a t?