In one of my problems, I have to integrate:
$$ \int_0^{2\pi} \vec{e}_\phi \: d\phi $$
What the solution to my problem seems to say is that because
$$ \vec{e}_\phi = cos\phi \, \vec{e}_x + sin\phi \, \vec{e}_y $$
I can't pull the unit vector out like a constant. I don't understand this.
The reason is that I could well argue that if what my solution does is right, then it must be true that
$$ \int \vec{e}_x \: dx \neq x\vec{e}_x $$
because in some coordinate system $\vec{e}_x$ is dependent of x:
$$ \vec{e}_x = f(x) \vec{e}_\phi $$
for example.
Intuitively, if I'm in cylindrical coordinates, the unit vectors have to be constant because a vector space should be able to exist on their own, without making any reference to another vector space.
Could someone help me understand why I have to take $\vec{e}_\phi$ as phi dependent when I integrate, although I am in cylindrical coordinates?