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:

• A. \(\dfrac{\pi}{4}\)
• B. \(\dfrac{\pi}{2}\)
• C. \(\dfrac{3\pi}{4}\)
• D. \(\pi\)

Hint:

For greatest integer function problems, first determine the range of the expression inside the bracket on the given interval.

Answer 1

The Correct Option is C

Step 1: Determine $ [4x/\pi] $ on $ [3\pi/4,\pi] $.
For $ x\in[3\pi/4,\pi) $: $ 4x/\pi\in[3,4) $, so $ [4x/\pi]=3 $.
Step 2: Find the range of $ \sin x $ on $ [3\pi/4,\pi] $.
$ \sin(3\pi/4)=\frac{1}{\sqrt{2}}\approx 0.707 $ and $ \sin\pi=0 $, so $ 0\leq\sin x\leq\frac{1}{\sqrt{2}}<1 $.
Step 3: Verify at sample points.
At $ x=3\pi/4 $: $ \sin x+3=3.707 \implies [\cdot]=3 $. At $ x=5\pi/6 $: $ 0.5+3=3.5 \implies [\cdot]=3 $. At $ x=11\pi/12 $: $ 0.259+3=3.259 \implies [\cdot]=3 $.
Step 4: Confirm the floor is 3 throughout.
Since $ 0\leq\sin x<1 $ and $ [4x/\pi]=3 $: $ 3\leq\sin x+3<4 \implies [\sin x+[4x/\pi]]=3 $ throughout the interval.
Step 5: Evaluate the integral.
\[ \int_{3\pi/4}^{\pi}3\,dx = 3\cdot\frac{\pi}{4}=\frac{3\pi}{4} \]
Step 6: State the answer.
\[ \boxed{\dfrac{3\pi}{4}} \]