limit of integral function As part of an investigation, a student of mine needed to evaluate the limit, $$\lim_{\beta\to 0+}\int_0^\beta\frac{1}{\sqrt{\cos\theta-\cos\beta}}\,d\theta.$$ Mathematica gives the answer as $\frac{\pi}{\sqrt2}$, but I couldn't show her a nice way of proving this. Is there any way to do this that a (bright) high school student might understand? Or does it necessarily involve an excursion into the land of elliptic integrals? For interest's sake: the limit arose when she was trying to see how well the circle approximates the tautochrone.

if you substitute $\theta = \beta t$, you get \begin{equation} \int_0^1 dt \frac{\beta}{\sqrt{\cos\beta t - \cos\beta}} \end{equation} and you do a Taylor expansion of the integrand in $\beta$ (essentially take the limit $\beta$ goes to $0$).

You are left with $\int_0^1 dt\sqrt{2}\frac{1}{\sqrt{1-t^2}} $, which is your result.