Question

${\int }_0^{\pi }\left[\cot x\right]$ dx, where $[.]$ denotes the greatest integer function, is equal to

• A. $\pi /2$
• B. $1$
• C. $-1$
• D. $-\pi /2$

Answer 1

The correct option is D $-\pi /2$

Let $I={\int }_0^{\pi }\left[\cot x\right]$ dx ... (i)
Then, also $I={\int }_0^{\pi }\left[\cot (\pi -x)\right]dx={\int }_0^{\pi }[-\cot x]dx$ ... (ii)
From (i) & (ii) eqs, we get
$2I={\int }_0^{\pi }\left\{\right[\cot x]+[-\cot x\left]\right\}dx$
$={\int }_0^{\pi }(-1)dx=-\pi$
$\Rightarrow I={\int }_0^{\pi }(-1)dx=-\frac{\pi }{2}$
Hence, option 'D' is correct.