Robert J. did great work.
For John let me explain more detail processes.
x = t cost dx/dt = cost - tsint
y = t sint dy/dt = sint + tcost
(dβ/dt)2 = (dx/dt)2 + (dy/dt)2 ; in here β is a arc length and dβ is a small segment of this
dβ/dt = √((dx/dt)2 + (dy/dt)2 )
= √( ( cost^2 -2tcost sint + t2 sint^2 ) + (sint^2 +2tsint cost +t2cost^2) )
= √( (cost^2 + sint^2) + t2(sint^2 +cost^2) )
= √( 1 + t2 )
β = ∫(from -1 to 1) √( 1 + t2 ) dt
to solve above equation, let t = tanx , ( this is an easier way to solve this type's of problem. If you solve many problems you will know it by experiences )
t = tanx dt = secx^2 dx (and when t = 1, x = inverse tan 1, therefore x = pi/4)
so, when t varies from -1 to 1 it is better to calculate 2 times of integration of x from 0 to pi/4.
therefore, β = 2∫(from 0 to pi/4) √(1+tant^2) secx^2 dx = 2∫(from 0 to pi/4) √( secx^3 ) dx
after this step, you have to do the partial integration. (this process is also hard).
The result of Robert J. is correct.