Question:
If \([\,\cdot\,]\) represents greatest integer function, then
\[
\int_{\frac{3\pi}{4}}^{\pi}
\left[\sin x+\left[\frac{4x}{\pi}\right]\right]\,dx
=
\]
is equal to:
If \([\,\cdot\,]\) represents greatest integer function, then
\[
\int_{\frac{3\pi}{4}}^{\pi}
\left[\sin x+\left[\frac{4x}{\pi}\right]\right]\,dx
=
\]
is equal to:
Show Hint
For greatest integer function problems, first determine the range of the expression inside the bracket on the given interval.
Updated On: Jun 25, 2026
- \(\dfrac{\pi}{4}\)
- \(\dfrac{\pi}{2}\)
- \(\dfrac{3\pi}{4}\)
- \(\pi\)
Show Solution
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The Correct Option is C
Solution and Explanation
Step 1: Understand the interval.
Given interval is \[ \frac{3\pi}{4}\leq x\leq \pi \] Multiply by \(\frac{4}{\pi}\): \[ 3\leq \frac{4x}{\pi}\leq 4 \] Therefore, \[ \left[\frac{4x}{\pi}\right]=3 \] for \[ \frac{3\pi}{4}\leq x\lt \pi \] At \(x=\pi\), the value becomes \(4\), but a single point does not affect the value of the definite integral.
Step 2: Analyze \(\sin x\) on the interval.
For \[ x\in \left[\frac{3\pi}{4},\pi\right], \] we have \[ 0\leq \sin x\leq \frac{1}{\sqrt{2}} \] Hence, \[ 3\leq 3+\sin x\lt 4 \] Therefore, \[ \left[3+\sin x\right]=3 \] So, \[ \left[\sin x+\left[\frac{4x}{\pi}\right]\right]=3 \]
Step 3: Evaluate the integral.
Thus, \[ \int_{\frac{3\pi}{4}}^{\pi} \left[\sin x+\left[\frac{4x}{\pi}\right]\right]\,dx = \int_{\frac{3\pi}{4}}^{\pi}3\,dx \] \[ = 3\left[x\right]_{\frac{3\pi}{4}}^{\pi} \] \[ = 3\left(\pi-\frac{3\pi}{4}\right) \] \[ = 3\cdot \frac{\pi}{4} \] \[ = \frac{3\pi}{4} \]
Step 4: Final conclusion.
Therefore, \[ \boxed{\frac{3\pi}{4}} \]
Given interval is \[ \frac{3\pi}{4}\leq x\leq \pi \] Multiply by \(\frac{4}{\pi}\): \[ 3\leq \frac{4x}{\pi}\leq 4 \] Therefore, \[ \left[\frac{4x}{\pi}\right]=3 \] for \[ \frac{3\pi}{4}\leq x\lt \pi \] At \(x=\pi\), the value becomes \(4\), but a single point does not affect the value of the definite integral.
Step 2: Analyze \(\sin x\) on the interval.
For \[ x\in \left[\frac{3\pi}{4},\pi\right], \] we have \[ 0\leq \sin x\leq \frac{1}{\sqrt{2}} \] Hence, \[ 3\leq 3+\sin x\lt 4 \] Therefore, \[ \left[3+\sin x\right]=3 \] So, \[ \left[\sin x+\left[\frac{4x}{\pi}\right]\right]=3 \]
Step 3: Evaluate the integral.
Thus, \[ \int_{\frac{3\pi}{4}}^{\pi} \left[\sin x+\left[\frac{4x}{\pi}\right]\right]\,dx = \int_{\frac{3\pi}{4}}^{\pi}3\,dx \] \[ = 3\left[x\right]_{\frac{3\pi}{4}}^{\pi} \] \[ = 3\left(\pi-\frac{3\pi}{4}\right) \] \[ = 3\cdot \frac{\pi}{4} \] \[ = \frac{3\pi}{4} \]
Step 4: Final conclusion.
Therefore, \[ \boxed{\frac{3\pi}{4}} \]
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