Explanation

(b)
(1)$I = \int_{- \pi/2}^{\pi/2}\sqrt{\cos{x - \cos^{3}x}}dx$
$= \int_{- \pi/2}^{\pi/2}\sqrt{\cos x\ }\sin{x\ dx}$
$= 2\int_{0}^{\pi/2}{\sqrt{\cos x}\sin{x\ dx}}$
$= - \frac{4}{3}\lbrack{\cos^{3/2}{x\rbrack}}_{0}^{\pi/2} = \frac{4}{3}$
(2)$I = \int_{0}^{1}{|x - 1|dx + \int_{1}^{4}{|x - 1|dx +}\int_{0}^{3}{|x - 3|dx + \int_{3}^{4}{|x - 3|dx}}}$
$= \int_{0}^{1}{- (x - 1)dx + \int_{1}^{4}{(x - 1)dx +}\int_{0}^{3}{- (x - 3)dx +}\int_{3}^{4}{(x - 3)dx}}$
$= 10$