Answer
$\dfrac{\pi}{4}$
Work Step by Step
Let us consider $x =\frac{1}{2} \sin \theta $
and $dx=\frac{1}{2} \cos \theta d \theta $
Now, the given integral becomes:
$\int \dfrac{ \cos \theta d \theta }{\sqrt{1-(1/4) \sin^2 \theta}} = \int d \theta$
Now, integrate with limits:
$\int_{0}^{1/2 \sqrt 2} d \theta=[sin^{-1} (2x)]_{0}^{1/2 \sqrt 2}=\dfrac{\pi}{4}$