Explanation

(b)
$\int_{}^{}\sqrt{1 + \sin\left( \frac{x}{4} \right)}\ dx = \int_{}^{}{\sqrt{1 + 2\ \sin\frac{x}{8}\cos\frac{x}{8}\ }dx}$
$= \int_{}^{}\sqrt{\sin^{2}\frac{x}{8} + \cos^{2}\frac{x}{8} + 2\sin\frac{x}{8}\cos\frac{x}{8}}\ dx$
$= \int_{}^{}\sqrt{\left( \sin{\frac{x}{8} + \cos\frac{x}{8}} \right)^{2}}\ dx$
$= \int_{}^{}\left( \sin\frac{x}{8} + \cos\frac{x}{8} \right)\ dx$
$= \frac{- \cos\frac{x}{8}}{\frac{1}{8}} + \frac{\sin\frac{x}{8}}{\frac{1}{8}} + c$
$= 8\left( \sin\frac{x}{8} - \cos\frac{x}{8} \right) + c$