Explanation

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