Question
Let $\lbrack \cdot \rbrack$ denote the greatest integer function. Then $\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}}\mspace{2mu}\left( \frac{12(3 + \lbrack x\rbrack)}{3 + \lbrack sinx\rbrack + \lbrack cosx\rbrack} \right)dx$ is equal to:
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Solution
$I = \int_{- \frac{\pi}{2}}^{\frac{\pi}{2}}\mspace{2mu}\frac{12(3 + \lbrack x\rbrack)dx}{3 + \lbrack sinx\rbrack + \lbrack cosx\rbrack}$
$${I = \int_{- \frac{\pi}{2}}^{- 1}\mspace{2mu}\frac{12(1)dx}{2} + \int_{- 1}^{0}\mspace{2mu}\frac{12(2)dx}{2} + \int_{0}^{1}\mspace{2mu}\frac{12(3)dx}{3} + \int_{1}^{\frac{\pi}{2}}\mspace{2mu}\frac{12(4)dx}{3} }{I = 6\left( \frac{\pi}{2} - 1 \right) + 12(0 + 1) + 12(1 - 0) + 16\left( \frac{\pi}{2} - 1 \right) }{I = 3\pi - 6 + 12 + 12 + 8\pi - 16 }{I = 11\pi + 2}$$
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