Explanation

(b)
Let $I = \int_{- 3\pi/2}^{- \pi/2}{\lbrack(x + \pi)^{3}} + \cos^{2}{x\rbrack dx}$ …(i)
$\Rightarrow I = \int_{- 3\pi/2}^{- \pi/2}\left\lbrack \left( - \frac{\pi}{2} - \frac{3\pi}{2} - x + \pi \right)^{3} \right.\$
$= \cos^{2}\left. \ \left( - \frac{\pi}{2} - \frac{3\pi}{2} - x \right) \right\rbrack dx$
$\Rightarrow I = \int_{- 3\pi/2}^{- \pi/2}{\lbrack - (x + \pi)^{3} + \cos^{2}{\rbrack dx}\ldots(ii)}$
On adding Eqs. (i)and (ii),we get
$2I = \int_{- 3\pi/2}^{- \pi/2}{2\cos^{2}{x\ dx}}$
$= \int_{- 3\pi/2}^{- \pi/2}{(1 + \cos{2x)dx}}$
$= \left\lbrack x + \frac{\sin{2x}}{2} \right\rbrack_{- 3\pi/2}^{- \pi/2}$
$= \left\lbrack - \frac{\pi}{2} + \frac{\sin{( - \pi)}}{2} - \left( - \frac{3\pi}{2} + \frac{sin( - 3\pi)}{2} \right) \right\rbrack = \pi$
$\Rightarrow I = \frac{\pi}{2}$