Question

The value of ${\int }_0^{100}\left[{\tan }^{-1}x\right]dx$ is (where [.] is G.I.F.)

• A. $100$
• B. $100-{\tan }^{-1}1$
• C. $100-\tan 1$
• D. $100+\frac{\pi }{4}$

Answer 1

The correct option is C $100-\tan 1$

$I={\int }_0^{100}\left[{tan}^{-1}x\right]dx={\int }_0^{tan1}\left[{tan}^{-1}x\right]dx+{\int }_{tan1}^{100}\left[{tan}^{-1}x\right]dx$

For $0$

$\Rightarrow \left[{tan}^{-1}x\right]=0$ ...(1)

And for $tan1$

$\Rightarrow \left[{tan}^{-1}x\right]=1$ ...(2)

From (1) and (2), we get

$I={\int }_0^{tan1}0.dx+{\int }_{tan1}^{100}1.dx=0+{x|}_{tan1}^{100}=100-tan1$