Question

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

• A. $\dfrac{1}{60}(\pi-7)$
• B. $\dfrac{1}{60}(21\pi-1)$
• C. $\dfrac{7}{60}(3\pi-1)$
• D. $\dfrac{7}{60}(\pi-3)$

Hint:

For integrals involving the greatest integer function, always split the interval at integer points and integrate piecewise.

Answer 1

The Correct Option is C

To solve the integral 

\[\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1}{[x]+4}\,dx\]

 where \([x]\) denotes the greatest integer function, we need to analyze the behavior of the function inside the integral.

1. First, understand the range of \(x\) and how the greatest integer function \([x]\) behaves. The greatest integer function, \([x]\), returns the greatest integer less than or equal to \(x\).
2. The interval of integration is \([-\frac{\pi}{2}, \frac{\pi}{2}]\). We approximate \(\pi \approx 3.14\), so \(-\frac{\pi}{2} \approx -1.57\) and \(\frac{\pi}{2} \approx 1.57\).
3. Analyze \([x]\) in this interval:
   • For \(x \in [-\frac{\pi}{2}, -1)\), \([x] = -2\).
   • For \(x \in [-1, 0)\), \([x] = -1\).
   • For \(x \in [0, 1)\), \([x] = 0\).
   • For \(x \in [1, \frac{\pi}{2}]\), \([x] = 1\).
4. Split the integral based on these intervals:
5. Calculate each integral separately:
   • \(\int_{-\frac{\pi}{2}}^{-1} \frac{1}{2}\,dx = \frac{1}{2}\left([-1] - \left(-\frac{\pi}{2}\right)\right) = \frac{1}{2}(1 + \frac{\pi}{2})\)
   • \(\int_{-1}^{0} \frac{1}{3}\,dx = \frac{1}{3}[0 - (-1)] = \frac{1}{3}\)
   • \(\int_{0}^{1} \frac{1}{4}\,dx = \frac{1}{4}[1 - 0] = \frac{1}{4}\)
   • \(\int_{1}^{\frac{\pi}{2}} \frac{1}{5}\,dx = \frac{1}{5}(\frac{\pi}{2} - 1)\)
6. Add these results together:
7. Combine and simplify:
8. By further simplification, combine terms with \(\pi\) and constant terms:

Therefore, the value of the integral is \(\frac{7}{60} (3\pi - 1)\), which corresponds to the correct answer. Thus, the correct answer is:

The value of the integral is: \(\frac{7}{60} (3\pi - 1)\)