Question

$I = \int_{- \pi/2}^{\pi/2}\mspace{2mu}\frac{1}{\lbrack x\rbrack + 4}dx$

$${I = \int_{- \pi/2}^{- 1}\mspace{2mu}\frac{dx}{2} + \int_{- 1}^{0}\mspace{2mu}\frac{dx}{3} + \int_{1}^{0}\mspace{2mu}\frac{dx}{4} + \int_{1}^{\pi/2}\mspace{2mu}\frac{dx}{5} }{= \frac{1}{2}\left( - 1 + \frac{\pi}{2} \right) + \frac{1}{3}(1) + \frac{1}{4}(1) + \left( \frac{\pi}{2} - 1 \right)\frac{1}{5} }{= \frac{7\pi}{20} - \frac{7}{60} = \frac{7}{60}(3\pi - 1)}$$