I'm having trouble finding the constant of integration in parametric, vector-based equations.
Given an equation:
$$ a(t)\ =\langle \cos(t),\ \sin(t)\rangle $$
and
$$ \int\ a(t)\ dt\ =\langle 0,\ 2\rangle $$
for some $t$, I need to calculate the constant of integration for the $i$ and $j$ components of $a(t)$.
So, I can get as far as finding the following:
$$ \int a(t)\ dt\ =\langle \sin(t), -\cos(t)\rangle + \mathbf C $$ $$ C_i = -\sin(t) $$ $$ C_j = 2 + \cos(t) $$
My hunch is I need to resolve t to some number--otherwise $C$ isn't a constant.
Any nudges in the right direction would be most appreciated.