Question

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

• A. $\frac{\pi }{2}$
• B. $1$
• C. $-1$
• D. $-\frac{\pi }{2}$

Answer 1

The correct option is D $-\frac{\pi }{2}$

Let $I={\int }_0^{\pi }\left[\cot x\right]dx$ ...(i)

As ${\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f(a-x)dx$

$\therefore I={\int }_0^{\pi }\left[\cot \right(\pi -x\left)\right]dx={\int }_0^{\pi }[-\cot x]dx$ ...(ii)

On adding Eqs. (i) and (ii), we get

$2I={\int }_0^{\pi }\left[\cot x\right]dx+{\int }_0^{\pi }[-\cot x]dx$

$={\int }_0^{\pi }(-1)dx$

$=[-x{]}_0^{\pi }=-\pi$

$\Rightarrow I=-\frac{\pi }{2}$