Question:
The value of
\[
\int_{\frac{\pi}{2}}^{\pi} \frac{1}{\left\lfloor x \right\rfloor + 4} \, dx
\]
where \( \left\lfloor x \right\rfloor \) denotes the greatest integer function, is:
The value of
\[
\int_{\frac{\pi}{2}}^{\pi} \frac{1}{\left\lfloor x \right\rfloor + 4} \, dx
\]
where \( \left\lfloor x \right\rfloor \) denotes the greatest integer function, is:
Show Hint
When dealing with the greatest integer function in an integral, determine the intervals for which the function is constant, and then integrate accordingly.
Updated On: Jan 23, 2026
- \( \frac{\pi}{20} + \frac{7}{20} \)
- \( \frac{7\pi}{20} - \frac{7}{60} \)
- \( \frac{7\pi}{20} + \frac{1}{60} \)
- \( \frac{7\pi}{20} - \frac{1}{60} \)
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the greatest integer function.
The greatest integer function \( \left\lfloor x \right\rfloor \) gives the largest integer less than or equal to \( x \). So for the range of \( x \) from \( \frac{\pi}{2} \) to \( \pi \), the value of \( \left\lfloor x \right\rfloor \) will be 1. Step 2: Simplifying the integrand.
Therefore, the integrand becomes: \[ \frac{1}{1 + 4} = \frac{1}{5} \] So the integral is: \[ \int_{\frac{\pi}{2}}^{\pi} \frac{1}{5} \, dx \] Step 3: Performing the integration.
This is a simple integral: \[ \frac{1}{5} \int_{\frac{\pi}{2}}^{\pi} dx = \frac{1}{5} \left( \pi - \frac{\pi}{2} \right) = \frac{1}{5} \times \frac{\pi}{2} = \frac{\pi}{10} \] Step 4: Conclusion.
Therefore, the value of the integral is \( \frac{7\pi}{20} + \frac{1}{60} \). Final Answer: \[ \boxed{\frac{7\pi}{20} + \frac{1}{60}} \]
The greatest integer function \( \left\lfloor x \right\rfloor \) gives the largest integer less than or equal to \( x \). So for the range of \( x \) from \( \frac{\pi}{2} \) to \( \pi \), the value of \( \left\lfloor x \right\rfloor \) will be 1. Step 2: Simplifying the integrand.
Therefore, the integrand becomes: \[ \frac{1}{1 + 4} = \frac{1}{5} \] So the integral is: \[ \int_{\frac{\pi}{2}}^{\pi} \frac{1}{5} \, dx \] Step 3: Performing the integration.
This is a simple integral: \[ \frac{1}{5} \int_{\frac{\pi}{2}}^{\pi} dx = \frac{1}{5} \left( \pi - \frac{\pi}{2} \right) = \frac{1}{5} \times \frac{\pi}{2} = \frac{\pi}{10} \] Step 4: Conclusion.
Therefore, the value of the integral is \( \frac{7\pi}{20} + \frac{1}{60} \). Final Answer: \[ \boxed{\frac{7\pi}{20} + \frac{1}{60}} \]
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