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

We are given the integral \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1}{[x]+4}\,dx \] where $[x]$ denotes the greatest integer less than or equal to $x$.
Step 1: Determine the values of $[x]$ in the interval.
Since \[ -\frac{\pi}{2}\approx -1.57 \quad \text{and} \quad \frac{\pi}{2}\approx 1.57, \] the interval $[-\frac{\pi}{2},\frac{\pi}{2}]$ can be divided as: \[ [-\tfrac{\pi}{2},-1),\ [-1,0),\ [0,1),\ [1,\tfrac{\pi}{2}] \] Step 2: Evaluate the integral piecewise.
\[ \int_{-\frac{\pi}{2}}^{-1} \frac{1}{3}\,dx + \int_{-1}^{0} \frac{1}{4}\,dx + \int_{0}^{1} \frac{1}{5}\,dx + \int_{1}^{\frac{\pi}{2}} \frac{1}{6}\,dx \] Step 3: Compute each integral.
\[ \int_{-\frac{\pi}{2}}^{-1} \frac{1}{3}\,dx = \frac{1}{3}\left(-1+\frac{\pi}{2}\right) \] \[ \int_{-1}^{0} \frac{1}{4}\,dx = \frac{1}{4} \] \[ \int_{0}^{1} \frac{1}{5}\,dx = \frac{1}{5} \] \[ \int_{1}^{\frac{\pi}{2}} \frac{1}{6}\,dx = \frac{1}{6}\left(\frac{\pi}{2}-1\right) \] Step 4: Add all the results.
\[ \frac{1}{3}\left(\frac{\pi}{2}-1\right) + \frac{1}{4} + \frac{1}{5} + \frac{1}{6}\left(\frac{\pi}{2}-1\right) \] \[ =\left(\frac{\pi}{2}-1\right)\left(\frac{1}{3}+\frac{1}{6}\right) +\frac{1}{4}+\frac{1}{5} \] \[ =\frac{\pi-2}{4}+\frac{9}{20} \] \[ =\frac{15\pi-5}{20} =\frac{7}{60}(3\pi-1) \] Final Answer: $\boxed{\dfrac{7}{60}(3\pi-1)}$