Let [t] denote the greatest integer

Q: Why is $[\csc x] = 1$ even though $\csc(\frac{\pi}{6}) = 2$?

The greatest integer function $[t]$ is defined as the greatest integer less than or equal to $t$. On the open interval $(\frac{\pi}{6}, \frac{5\pi}{6})$, the value of $\csc x$ is strictly between $1$ and $2$. Since the function never reaches $2$ inside this open interval, the greatest integer is always $1$.

Q: How do I handle the negative values of $\cot x$ when calculating the greatest integer?

Remember that for negative numbers, the greatest integer function rounds down to the next smaller integer. For example, if $\cot x = -0.5$, then $[\cot x] = -1$. Always draw a number line if you are confused!

Q: Why do we split the integral at $\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}$?

These are the points where $\cot x$ crosses integer values. Since the greatest integer function is a step function, it only changes its value at these specific points. By splitting the integral here, we turn a complex function into a series of simple constants.

Question

Let $[t]$ denote the greatest integer $\le t$. Then $\frac{2}{\pi} \int_{\pi/6}^{5\pi/6} (8[\csc x] - 5[\cot x]) \, dx$ is equal to

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are tasked with evaluating the integral:
$$I = \frac{2}{\pi} \int_{\pi/6}^{5\pi/6} (8[\csc x] - 5[\cot x]) \, dx$$

By the **linearity of the integral**, we can decompose this into two distinct parts:
$$I = \frac{16}{\pi} \int_{\pi/6}^{5\pi/6} [\csc x] \, dx - \frac{10}{\pi} \int_{\pi/6}^{5\pi/6} [\cot x] \, dx$$

Let us define these as $I_1$ and $I_2$ respectively.

### Phase 1: The Cosecant Mystery

Consider the function $f(x) = \csc x$ on the interval $(\frac{\pi}{6}, \frac{5\pi}{6})$. At the boundaries, $\csc(\frac{\pi}{6}) = 2$ and $\csc(\frac{5\pi}{6}) = 2$.

However, between these points, the function dips, reaching its minimum value of $1$ at $x = \frac{\pi}{2}$. Thus, for all $x$ in our open interval, $1 \le \csc x < 2$.

Because the function never reaches $2$ inside this interval, the greatest integer $[\csc x]$ is simply $1$. The integral $I_1$ becomes:
$$I_1 = \int_{\pi/6}^{5\pi/6} 1 \, dx = \frac{5\pi}{6} - \frac{\pi}{6} = \frac{4\pi}{6} = \frac{2\pi}{3}$$

### Phase 2: The Cotangent Odyssey

Now, we turn to $I_2 = \int_{\pi/6}^{5\pi/6} [\cot x] \, dx$. The cotangent function is strictly decreasing on this interval.

At $x = \frac{\pi}{6}$, $\cot x = \sqrt{3} \approx 1.732$. At $x = \frac{5\pi}{6}$, $\cot x = -\sqrt{3} \approx -1.732$.

As it descends, it crosses the integers $1, 0, -1$. We split the integral at the points where $\cot x$ equals these integers:
1. $\cot x = 1 \Rightarrow x = \frac{\pi}{4}$
2. $\cot x = 0 \Rightarrow x = \frac{\pi}{2}$
3. $\cot x = -1 \Rightarrow x = \frac{3\pi}{4}$

This divides our interval into four sub-intervals where the greatest integer is constant:
- $[\frac{\pi}{6}, \frac{\pi}{4}) \Rightarrow [\cot x] = 1$
- $[\frac{\pi}{4}, \frac{\pi}{2}) \Rightarrow [\cot x] = 0$
- $[\frac{\pi}{2}, \frac{3\pi}{4}) \Rightarrow [\cot x] = -1$
- $[\frac{3\pi}{4}, \frac{5\pi}{6}] \Rightarrow [\cot x] = -2$

### Phase 3: The Arithmetic Finale

We set up $I_2$ as the sum of these integrals:
$$I_2 = 1\left(\frac{\pi}{4} - \frac{\pi}{6}\right) + 0\left(\frac{\pi}{2} - \frac{\pi}{4}\right) + (-1)\left(\frac{3\pi}{4} - \frac{\pi}{2}\right) + (-2)\left(\frac{5\pi}{6} - \frac{3\pi}{4}\right)$$

Calculating these values:
$$I_2 = \frac{\pi}{12} + 0 - \frac{\pi}{4} - \frac{\pi}{6} = \frac{\pi - 3\pi - 2\pi}{12} = -\frac{4\pi}{12} = -\frac{\pi}{3}$$

Finally, we substitute $I_1 = \frac{2\pi}{3}$ and $I_2 = -\frac{\pi}{3}$ back into our original expression:
$$I = \frac{16}{\pi}\left(\frac{2\pi}{3}\right) - \frac{10}{\pi}\left(-\frac{\pi}{3}\right) = \frac{32}{3} + \frac{10}{3} = \frac{42}{3} = 14$$

The final result is **14**.

Animated Solution for Mathematics - Definite Integration: Let $[t]$ denote the greatest integer $\leq t$ . Then $\frac{2}{π} \int_{π /6}^{5 π /6} (8 [csc x] - 5 [cot x]) d x$ is equal to

Enter Numerical Value:

Visualized Solution

Splitting the Integral

Analyzing $[csc x]$

Greatest Integer of $csc x$

Evaluating $I_{1}$

Analyzing $[cot x]$

Splitting Points for $[cot x]$

Intervals for $[cot x]$

Setting up $I_{2}$

Evaluating $I_{2}$

Final Substitution

Final Answer

The Sigma Insight: Fundamental Theorem & Properties of Definite Integrals

Analyzing the Setup

Phase 1

Phase 2

Phase 3

Similar Questions

The value of $\int_{- π /2}^{π /2} \frac{d x}{[ x ] + [ s i n x ] + 4}$ , where $[t]$ denotes the greatest integer less than or equal to $t$ , is :

$\int_{0}^{π} [cot x] d x$ , where $[.]$ denotes the greatest integer function, is equal to: