Question

The value of ${\int }_0^{100}\left[{\tan }^{-1}x\right]dx$ is equal to ?
(Note: $[.]$ denotes the greatest integer function less than or equal to $x$)

• A. $\tan \left(1\right)-100$
• B. $\frac{\pi }{2}-\tan \left(1\right)$
• C. $100-\tan \left(1\right)$
• D. none of these

Answer 1

Answer: (C)

$100-\tan \left(1\right)$

$I={\int }_0^{100}\left[{\tan }^{-1}x\right]dx={\int }_0^{\tan 1}\left[{\tan }^{-1}x\right]dx+{\int }_{\tan 1}^{100}\left[{\tan }^{-1}x\right]dx$
For $x=0\Rightarrow \left[{\tan }^{-1}x\right]=0$ ...(1)
And for $x=\tan 1\Rightarrow \left[{\tan }^{-1}x\right]=1$ ...(2)
From (1) and (2), we get
$I={\int }_0^{\tan 1}0dx+{\int }_{\tan 1}^{100}1dx=0+{\left[x\right]}_{\tan 1}^{100}=100-\tan 1$