Math Help - Curvature problem

1. Curvature problem

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

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

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?

2. Just to get you going.

Let the curve be $c(t)=(x(t),y(t)), t\geq 0$. Suppose (without loss of generality) c is already parametrized by arc length. Then the formula $|c'(t)|=1$ gives $x'(t)^2+y'(t)^2=1$, so there exists a function $\phi(t)$ such that $x'(t)=\cos(\phi(t)), \ y'(t)=\sin(\phi(t))$. To explicitly calculate this function, recall that $k(t)=x'(t)y''(t)-x''(t)y'(t)$.