Question

Find the distance traveled by a particle with position (x, y) as t varies in the given time interval.
x = cos^2 t, y = cos t, 0 ≤ t ≤ 9π
and the length of the curve
so far i got
∫0 to 9pi √((-2cos(t)sin(t))^2+(-sint)^2)
I am having trouble integrating from here

Answer 1

4cos^2(t)sin^2(t) + sin^2(t)
= 4(1-sin^2(t))sin^2(t)+sin^2(t)
= 4sin^2(t)-4sin^2(t)+sin^2(t)
= 5sin^2(t)-4sin^4(t)
= sin^2(t) (5-4sin^2(t))
= sin^2(t) (1+4cos^2(t))

So, now you have
∫√(1+4cos^2(t)) sin(t) dt
u = cos(t)
du = -sin(t) dt

-∫√(1+4u^2) du
Now let
2u = tanθ

and see where that takes you.