Let k(t), $\displaystyle t \geq 0 $ construct the curve a(t), for which the curvature at $\displaystyle t \geq 0 $ is equal to k(t) and which satisfies the initial conditions a(0) = (0,0) and a'(0) = (1,0).

Prove that $\displaystyle a(t) = ( \int ^{t}_{0} cos ( \int ^{s}_{0}k(u)du)ds , \int ^{t}_{0} sin ( \int ^{s} _{0} k(u)du) ds ) $

Questions:

Now, if I substitute 0 in for t, I get (0,0). But taking the prime, should I take it relative to t or s?