The value of 2 2 ( 1 [ x ] + 4 ) d x , where [ · ] denotes the greatest integer function, is [2026]

Question

The value of $\int_{- \frac{π}{2}}^{\frac{π}{2}} (\frac{1}{[x] + 4}) d x,$ where $[\cdot]$ denotes the greatest integer function, is [2026]

Smart Achievers · Accepted Answer

(1)

$I = \int_{- π / 2}^{π / 2} \frac{1}{[x] + 4} d x$

$I = \int_{- π / 2}^{- 1} \frac{d x}{2} + \int_{- 1}^{0} \frac{d x}{3} + \int_{0}^{1} \frac{d x}{4} + \int_{1}^{π / 2} \frac{d x}{5}$

$= \frac{1}{2} (- 1 + \frac{π}{2}) + \frac{1}{3} (1) + \frac{1}{4} (1) + (\frac{π}{2} - 1) \frac{1}{5}$

$= \frac{7 π}{20} - \frac{7}{60} = \frac{7}{60} (3 π - 1)$

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