Instead of simplifying , you should have used and then the integral splits into three parts: integrating from 0 to , from to , and from to . The answer 8 is correct.
This is actually an arc length problem for the polar equation r=1+sin(x) (x being theta) After some work you get:
L=integral from 0 to 2 times pi (2+2*sin(x))^(1/2) dx. It gets reduced to
L=2^(1/2) times integral from 0 to 2 times pi (cos(x)/(1-sin(x))^1/2) dx
But if u=1-sin(x) don't the limits of integration both equal 1?
The book says the answer is L=8 The figure is a cardioid.