Integral Calculation with Greatest Integer Function
Evaluate the definite integral $ I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1}{[x]+4} dx $. The limits are approximately $[-1.57, 1.57]$.
Step 1: Determine Intervals for $[x]$
The greatest integer function $[x]$ changes value at integers. Within $[-1.57, 1.57]$, the integers are -1, 0, 1. We split the integral based on these values:
- Interval $[-\frac{\pi}{2}, -1)$: $[x] = -2$. Length $\frac{\pi}{2} - 1$.
- Interval $[-1, 0)$: $[x] = -1$. Length $1$.
- Interval $[0, 1)$: $[x] = 0$. Length $1$.
- Interval $[1, \frac{\pi}{2}]$: $[x] = 1$. Length $\frac{\pi}{2} - 1$.
Step 2: Compute Integral Over Each Interval
Calculate the integral for each sub-interval:
- For $[-\frac{\pi}{2}, -1)$: Integral = $\int_{-\frac{\pi}{2}}^{-1} \frac{1}{-2+4} dx = \frac{1}{2} \times (\frac{\pi}{2} - 1)$.
- For $[-1, 0)$: Integral = $\int_{-1}^{0} \frac{1}{-1+4} dx = \frac{1}{3} \times 1 = \frac{1}{3}$.
- For $[0, 1)$: Integral = $\int_{0}^{1} \frac{1}{0+4} dx = \frac{1}{4} \times 1 = \frac{1}{4}$.
- For $[1, \frac{\pi}{2}]$: Integral = $\int_{1}^{\frac{\pi}{2}} \frac{1}{1+4} dx = \frac{1}{5} \times (\frac{\pi}{2} - 1)$.
Step 3: Sum the Results
The total integral is the sum of the contributions:
$ I = \frac{1}{2}\left(\frac{\pi}{2} - 1\right) + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}\left(\frac{\pi}{2} - 1\right) $
The value obtained from this sum corresponds to Option A.
Final Answer: The final answer is $\boxed{\text{Option A}}$
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