Question

The value of the integral $\int_π/2³π/2[sin~x]dx$, where $[·]$ denotes the greatest integer function, is

• A. $\frac{\pi}{2}$
• B. $-\frac{\pi}{2}$
• C. 0
• D. $\pi$

Hint:

The value of the integral $\intπ/2[sin~x]dx$, where $[·]$ denotes the greatest integer function, is

Answer 1

The Correct Option is B

Step 1: Concept
Split the integral at points where the integer value of $\sin x$ changes.
Step 2: Analysis
Interval 1: $x \in [\pi/2, \pi]$, $\sin x$ is between 0 and 1, so $[\sin x] = 0$. Interval 2: $x \in (\pi, 3\pi/2]$, $\sin x$ is between -1 and 0, so $[\sin x] = -1$.
Step 3: Evaluation
Integral $= \int_{\pi/2}^{\pi} 0 \, dx + \int_{\pi}^{3\pi/2} -1 \, dx$.
Step 4: Conclusion
$= 0 + [-x]_{\pi}^{3\pi/2} = -(3\pi/2 - \pi) = -\pi/2$.
Final Answer: (b)