Question

The value of $\int\limits^{\pi/2}_{-\pi/2} \frac{dx}{\left[x\right]+\left[\sin x\right]+4} $ where [t] denotes the greatest integer less than or equal to t, is :

• A. $\frac{1}{12} (7 \pi + 5)$
• B. $\frac{3}{10} (4\pi - 3)$
• C. $\frac{1}{12} (7\pi - 5)$
• D. $\frac{3}{20} (4\pi - 3)$

Answer 1

The Correct Option is D

 To solve the integral \(\int\limits^{\pi/2}_{-\pi/2} \frac{dx}{\left[x\right]+\left[\sin x\right]+4}\), we need to evaluate how the greatest integer function affects the integrand over the given interval and compute the resultant sum.

1. Start by considering the greatest integer function, denoted \([x]\) and \([\sin x]\). The expression \([x] + [\sin x]\) changes based on the values of \(x\) in the interval \([-\pi/2, \pi/2]\).
2. Split the integral into segments where \(\left[x\right]\) is constant. This means focusing on integer intervals within the limit:
   • For \(x \in [-\pi/2, 0)\), \(\left[x\right] = -1\).
   • For \(x \in [0, \pi/2)\), \(\left[x\right] = 0\).
3. Next, consider \([\sin x]\). For \(x\) in \([-\pi/2, \pi/2]\), the range of \(\sin x\) is \([-1, 1]\). Thus, \([\sin x]\) can be \(-1$, $0$, or $1\):
   • For \(x\) in \([-\pi/2, 0]\), \([\sin x] = -1\), since \(\sin x \leq 0\).
   • For \(x\) close to zero but non-positive, \([\sin x] = -1\).
   • For \(x\) in \((0, \pi/2)\), \([\sin x] = 0\), since \(\sin x\) is positive.
4. Now, substitute back into the integrand: \(\left[x\right]+\left[\sin x\right]+4\) becomes:
   • \([-1] + [-1] + 4 = 2\) for \(x \in [-\pi/2, 0)\).
   • \([0] + [0] + 4 = 4\) for \(x \in [0, \pi/2]\).
5. Breaking the integral into two parts:
   • \(\int\limits_{-\pi/2}^{0} \frac{dx}{2} = \left[ \frac{x}{2} \right]_{-\pi/2}^{0} = \frac{0 - (-\pi/2)}{2} = \frac{\pi}{4}\)
   • \(\int\limits_{0}^{\pi/2} \frac{dx}{4} = \left[ \frac{x}{4} \right]_{0}^{\pi/2} = \frac{\pi/2 - 0}{4} = \frac{\pi}{8}\)
6. Add the results of these two integrals: \(\frac{\pi}{4} + \frac{\pi}{8} = \frac{2\pi}{8} + \frac{\pi}{8} = \frac{3\pi}{8}\).

Comparing with the given options, the correct answer is \(\frac{3}{20} (4\pi - 3)\).