Explanation

(b)
$\because\sin^{- 1}{(\cos{x) + \cos^{- 1}{(\sin{x)}}}}$ is periodic with period $2\pi$, then
$\int_{a + 2\pi}^{a + 5\pi/2}{(\sin^{- 1}{(\cos{x) + \cos^{- 1}{(\sin{x))\ dx}}}}}$
$= \int_{0}^{\pi/2}\left( \sin^{- 1}{\left\{ \cos\left( \frac{\pi}{2} - x \right) \right\} + \cos^{- 1}\left\{ \sin\left( \frac{\pi}{2} - x \right) \right\}} \right)dx$
$= \int_{0}^{\pi/2}{(\sin^{- 1}{\sin{x + \cos^{- 1}{\cos x)dx}}}}$
$= 2\int_{0}^{\pi/2}{x\ dx = 2\left\{ \frac{x^{2}}{2} \right\}_{0}^{\frac{\pi}{2}} = \frac{\pi^{2}}{4}}$