Question

The value of ${\int }_{-\tan 1}^0[-{\tan }^{-1}x]dx$, where $[.]$ represent greatest integer function is

• A. $1$
• B. $2$
• C. $3$
• D. $0$

Answer 1

The correct option is D $0$

$f\left(x\right)=ta{n}^{-1}x$ takes value $(-\frac{\pi }{2},0)\forall x\in [-\infty ,0]$

$\therefore g\left(x\right)=-ta{n}^{-1}x$ takes values $(0,\frac{\pi }{2})$

$\forall x\in (-ta{n}^1,0),g\left(x\right)\in (0,1)$

$\Rightarrow \left[g\right(x\left)\right]=[-ta{n}^{-1}x]=0$

$I={\int }_{tan1}^0[-ta{n}^{-1}x]dx=0$

$\therefore$ Option D is correct