Answer
$$\frac{{\sqrt 2 }}{4}$$
Work Step by Step
$$\eqalign{
& \int_0^{\pi /8} {\sqrt {1 - \cos 8x} dx} \cr
& = \int_0^{\pi /8} {\sqrt {1 - \cos 2\left( {4x} \right)} dx} \cr
& {\text{Use the identity }}\cos 2\theta = 1 - 2{\sin ^2}\theta \cr
& \int_0^{\pi /8} {\sqrt {1 - \cos 2\left( {4x} \right)} dx} = \int_0^{\pi /8} {\sqrt {1 - 1 + 2{{\sin }^2}4x} dx} \cr
& = \int_0^{\pi /8} {\sqrt {2{{\sin }^2}4x} dx} \cr
& = \sqrt 2 \int_0^{\pi /8} {\sin 4xdx} \cr
& {\text{Integrate}} \cr
& = \sqrt 2 \left[ { - \frac{1}{4}\cos 4x} \right]_0^{\pi /8} \cr
& = - \frac{{\sqrt 2 }}{4}\left[ {\cos \left( {\frac{\pi }{2}} \right) - \cos \left( 0 \right)} \right] \cr
& = \frac{{\sqrt 2 }}{4} \cr} $$
