Find the arc length of the curve

Question

𝑥(𝑡)=cos𝑡+𝑡sin𝑡, 0≤𝑡≤𝜋/2
𝑦(𝑡) = sin 𝑡 − 𝑡 cos 𝑡 ^2

Answer 1

(ds)^2 = (dx/dt)^2 + (dy/dt)^2
= (-sint + sint + tcost)^2 + (cost - cost + tsint)^2
= t^2cos^2(t) + t^2sin^2(t)
= t^2
so,

ds = dt

that's pretty easy to integrate, right?

Hmmm. I missed that ^2 hanging out there. Is

y(t) = sint - t(cost)^2
or
y(t) = sint - tcos(t^2)?

In either case, adjust the expression for ds above.