I am trying to get a two-terms asymptotic expansion {as $c\to 0 $ } for the following integral
$$\int_{0}^{c} \sqrt{\frac{c^2-\zeta^2}{1-\zeta^2}} d\zeta$$
I have tried so far to substitute $\zeta=c\ sin(\theta)$ and got the following form $$c^2\int_{0}^{\pi/2} \frac{cos^2\theta}{\sqrt{1-c^2sin^2\theta}} d\theta$$
how do I proceed from here?