Let [t] denote the greatest integer t.

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 \_\_\_\_\_\_\_\_\_\_.

Answer: 14

Accepted Answer

On $[\pi/6,5\pi/6]$, $\csc x\ge1$ so $[\csc x]=1$ throughout (equal to 1 except possibly at $x=\pi/2$ where it's exactly 1 too):

$$8\int_{\pi/6}^{5\pi/6}[\csc x]\,dx = 8\int_{\pi/6}^{5\pi/6}1\,dx = 8\cdot\frac{2\pi}3 = \frac{16\pi}3$$

For $\cot x$: by King's rule ($x\to\pi-x$), $[\cot x]+[-\cot x]=-1$ (since $\cot x$ is never an integer on this range except possibly $x=\pi/2$), so

$$I=\int_{\pi/6}^{5\pi/6}[\cot x]\,dx,\qquad 2I=\int_{\pi/6}^{5\pi/6}\big([\cot x]+[-\cot x]\big)dx=-\int_{\pi/6}^{5\pi/6}dx=-\frac{2\pi}3$$

$$I=-\frac{\pi}3 \implies -5I = \frac{5\pi}3$$

$$\frac2\pi\left(\frac{16\pi}3+\frac{5\pi}3\right) = \frac2\pi\cdot\frac{21\pi}3 = 14$$

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