Question

${\int }_0^{\mathrm{\pi }}\left[cotx\right]dx$, $\left[.\right]$ denotes the greatest integer function, that is :

• A. $\mathrm{\pi }/2$
• B. $1$
• C. $-1$
• D. $-\left(\mathrm{\pi }/2\right)$

Answer 1

The correct option is D

$-\left(\mathrm{\pi }/2\right)$

Evaluate the given integral.

Let the integral be : $I={\int }_0^{\mathrm{\pi }}\left[cotx\right]dx......\left(1\right)$

We know that the property of definite integral ${\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f\left(a-x\right)dx$

Express the Equation $\left(1\right)$ by using the above property

$$\begin{gathered} I={\int }_0^{\mathrm{\pi }}\left[cot\left(\mathrm{\pi }-\mathrm{x}\right)\right]dx \\ I={\int }_0^{\mathrm{\pi }}\left[-cot\mathrm{x}\right]dx......\left(2\right) \end{gathered}$$

We know that,

$$\begin{gathered} \left[x\right]+\left[-x\right]=-1,x\notin z \\ \left[x\right]+\left[-x\right]=0,x\in z \end{gathered}$$

Adding the Equation $\left(1\right)\text{and}\left(2\right)$

$$\begin{gathered} 2I={\int }_0^{\mathrm{\pi }}\left[cotx\right]dx+{\int }_0^{\mathrm{\pi }}\left[-cotx\right]dx \\ 2I={\int }_0^{\mathrm{\pi }}-1dx \\ 2I=\left(-1\right){\int }_0^{\mathrm{\pi }}dx \\ 2I=\left(-1\right){\left[x\right]}_0^{\mathrm{\pi }} \\ 2\mathrm{I}=\left(-1\right)\left[\mathrm{\pi }-0\right] \\ \mathrm{I}=-\frac{\mathrm{\pi }}{2} \end{gathered}$$

Hence, the correct answer is option D.