Question

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

• A. \( \dfrac{\pi}{20}+\dfrac{7}{20} \)
• B. \( \dfrac{7\pi}{20}-\dfrac{7}{60} \)
• C. \( \dfrac{7\pi}{20}-\dfrac{1}{60} \)
• D. \( \dfrac{7\pi}{20}+\dfrac{1}{60} \)

Hint:

For integrals involving \([x]\):
Always split the interval at integers
Replace \([x]\) by its constant value on each subinterval
Integrate piecewise and add the results

Answer 1

The Correct Option is D

Concept: When an integral contains the greatest integer function \([x]\), the interval of integration must be divided into subintervals over which \([x]\) has a constant value. Within each such subinterval, the integrand becomes a constant.
Step 1: Identify the values of \([x]\) over the interval \[ \left[\frac{\pi}{2},\,\pi\right] \] Numerically, \[ \frac{\pi}{2}\approx 1.57,\quad \pi\approx 3.14 \] Therefore:
For \(x\in\left[\frac{\pi}{2},2\right)\), \([x]=1\)
For \(x\in[2,3)\), \([x]=2\)
For \(x\in[3,\pi]\), \([x]=3\)
Step 2: Break the integral into corresponding subintervals. \[ \int_{\frac{\pi}{2}}^{\pi} \frac{dx}{[x]+4} = \int_{\frac{\pi}{2}}^{2} \frac{dx}{1+4} +\int_{2}^{3} \frac{dx}{2+4} +\int_{3}^{\pi} \frac{dx}{3+4} \]
Step 3: Evaluate each integral separately. \[ \int_{\frac{\pi}{2}}^{2} \frac{dx}{5} = \frac{1}{5}\left(2-\frac{\pi}{2}\right) \] \[ \int_{2}^{3} \frac{dx}{6} = \frac{1}{6}(1) = \frac{1}{6} \] \[ \int_{3}^{\pi} \frac{dx}{7} = \frac{1}{7}(\pi-3) \]
Step 4: Add all the evaluated parts. \[ \frac{1}{5}\left(2-\frac{\pi}{2}\right) + \frac{1}{6} + \frac{1}{7}(\pi-3) \] Simplifying, \[ = \left(\frac{2}{5}-\frac{\pi}{10}\right) + \frac{1}{6} + \left(\frac{\pi}{7}-\frac{3}{7}\right) \] \[ = \left(\frac{\pi}{7}-\frac{\pi}{10}\right) + \left(\frac{2}{5}+\frac{1}{6}-\frac{3}{7}\right) \] \[ = \frac{7\pi}{20}+\frac{1}{60} \]