Question

Let [t] denote the greatest integer ≤t. Then \(\frac{2}{π} \int\limits_{\frac{\pi}{6}}^{\frac{5\pi}{6}} (8 [cosec x]-5[cot x]) dx\)is equal to ______.

Hint:

To handle piecewise integrals, split the interval into regions where the floor functions remain constant.

Answer 1

Correct Answer: 14

We are given the integral: \[ \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} \left(8 [\csc x]\right) dx. \] Step 1: Evaluate the First Integral.
We first evaluate the integral involving \([\csc x]\): \[ \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} 8 [\csc x] dx. \] On the interval \(\left[\frac{\pi}{6}, \frac{5\pi}{6}\right]\), the values of \([\csc x]\) are as follows: - From \(\frac{\pi}{6}\) to \(\frac{\pi}{2}\), \([\csc x] = 2\), - From \(\frac{\pi}{2}\) to \(\frac{5\pi}{6}\), \([\csc x] = 1\). Thus, the integral can be split into two parts: \[ 8 \int_{\frac{\pi}{6}}^{\frac{\pi}{2}} 2 \, dx + 8 \int_{\frac{\pi}{2}}^{\frac{5\pi}{6}} 1 \, dx. \] Evaluating each part: \[ 8 \times 2 \left( \frac{\pi}{2} - \frac{\pi}{6} \right) + 8 \left( \frac{5\pi}{6} - \frac{\pi}{2} \right) = 16 \times \frac{\pi}{3} + 8 \times \frac{\pi}{3} = \frac{16\pi}{3} + \frac{8\pi}{3} = \frac{24\pi}{3} = 8\pi. \] Step 2: Evaluate the Integral Involving \([\cot x]\).
Now, we evaluate the second integral: \[ \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} [\cot x] dx. \] Since \(\cot x\) changes sign over the interval, we use the substitution \( x \rightarrow \pi - x \) to simplify the process. This results in: \[ \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} [\cot x] dx = \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} [- \cot x] dx. \] Thus, we now have: \[ 2I = \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} \left( \cot x + (-\cot x) \right) dx. \] This simplifies to: \[ 2I = \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} 0 \, dx. \] Thus, the integral evaluates to: \[ I = -\frac{1}{2} \int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} dx = -\frac{1}{2} \left( \frac{4\pi}{6} \right) = -\frac{\pi}{3}. \] Step 3: Combine the Results.
Now, combining the results from both integrals: \[ 2 \left( \frac{16\pi}{3} + \frac{5\pi}{3} \right) = 2 \times \frac{21\pi}{3} = 14. \] Thus, the value of the integral is: \[ \boxed{14}. \]