The value of int_{-frac{pi}{2}}^{frac{pi}{2}} | vedclass

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(C) Let $I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1}{[x]+4} dx$. Since $-\frac{\pi}{2} \approx -1.57$ and $\frac{\pi}{2} \approx 1.57$, we spli

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$\frac{7}{60}(3\pi-1)$

Vedclass · Answer

(C) Let $I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1}{[x]+4} dx$. Since $-\frac{\pi}{2} \approx -1.57$ and $\frac{\pi}{2} \approx 1.57$, we split the integral based on the values of $[x]$: $I = \int_{-\frac{\pi}{2}}^{-1} \frac{1}{[x]+4} dx + \int_{-1}^{0} \frac{1}{[x]+4} dx + \int_{0}^{1} \frac{1}{[x]+4} dx + \int_{1}^{\frac{\pi}{2}} \frac{1}{[x]+4} dx$ For $x \in [-\frac{\pi}{2}, -1)$, $[x] = -2$. For $x \in [-1, 0)$, $[x] = -1$. For $x \in [0, 1)$, $[x] = 0$. For $x \in [1, \frac{\pi}{2}]$, $[x] = 1$. $I = \int_{-\frac{\pi}{2}}^{-1} \frac{1}{-2+4} dx + \int_{-1}^{0} \frac{1}{-1+4} dx + \int_{0}^{1} \frac{1}{0+4} dx + \int_{1}^{\frac{\pi}{2}} \frac{1}{1+4} dx$ $I = \frac{1}{2} [-1 - (-\frac{\pi}{2})] + \frac{1}{3} [0 - (-1)] + \frac{1}{4} [1 - 0] + \frac{1}{5} [\frac{\pi}{2} - 1]$ $I = \frac{1}{2} (\frac{\pi}{2} - 1) + \frac{1}{3} (1) + \frac{1}{4} (1) + \frac{1}{5} (\frac{\pi}{2} - 1)$ $I = \frac{\pi}{4} - \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{\pi}{10} - \frac{1}{5}$ $I = (\frac{\pi}{4} + \frac{\pi}{10}) + (-\frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{5})$ $I = \frac{7\pi}{20} + \frac{-30+20+15-12}{60} = \frac{7\pi}{20} - \frac{7}{60} = \frac{21\pi - 7}{60} = \frac{7}{60}(3\pi - 1)$

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1}{[x]+4} dx$, where $[\bullet]$ denotes the greatest integer function, is

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