A simple cha(lle)nge!

First thing to do is eliminate $\alpha$ from the interval of integration. Let us make the following substitution: $\theta = \alpha x$ . Then, we have

$\lim_{\alpha \to 0^+} \int_0^1 \dfrac{\alpha}{\sqrt{\cos(\alpha x) - \cos(\alpha)}}dx.$

Now we use the Dominated convergence theorem in order to exchange the limit with the integral. We can do so, because it is possible to bound the absolute value of the function by another integrable function, for example $\frac{\pi}{\sqrt{1-x^2}}$ . Then, $\begin{aligned} & \int_0^1 \lim_{\alpha \to 0^+} \dfrac{\alpha}{\sqrt{\cos(\alpha x) - \cos(\alpha)}}dx \\ = & \int_0^1 \dfrac{\sqrt{2}}{\sqrt{1-x^2}}dx \\ = & \dfrac{\sqrt{2}}{2} \pi. \end{aligned}$