These two definite integrals come from the brachistochrone problem, where $g$ denotes gravitational acceleration (9.81 $m/s^2$) and $C$ is a constant function. How am I supposed to solve them? I've tried a trigonometric substitution in the first one, but haven't gone too far.
1) $\mathcal{I}_1 = \displaystyle \frac{1}{\sqrt{2g}} \int_0^1 \sqrt\frac{4x^2 + 1}{x^2 + C} dx$
2) $\mathcal{I}_2 = \displaystyle \frac{1}{\sqrt{2g}} \int_0^1 \sqrt\frac{4x + 1}{4x(\sqrt{x} + C)} dx$