Explanation

(b)
Let $I = \int_{- \pi/2}^{\pi/2}\sqrt{\cos{x - \cos^{3}x}}dx$
$= \int_{- \pi/2}^{\pi/2}{\sqrt{\cos x}\ \left| \sin x \right|dx}$
$= 2\int_{0}^{\pi/2}{\sqrt{\cos x}\sin{xdx}}$
Put $\cos{x = t \Rightarrow - \sin{x\ dx = dt}}$
$\therefore I = - 2\int_{1}^{0}{\sqrt{t}dt = - \frac{4}{3}\lbrack t^{3/2}\left. \ \ \right\rbrack_{1}^{0} = \frac{4}{3}}$