$\underset{0}{\overset{102}{\int }}\left[{\tan }^{-1}x\right]dx=$  (Where $\left[x\right]$  is the largest integer not exceeding $x$ )

for  $0\le x\le \tan 1,$
$0\le {\tan }^{-1}x\le 1$ and  $\tan 1\le x\le 102$
$1\le {\tan }^{-1}<2$
Hence $\underset{0}{\overset{102}{\int }}\left[{\tan }^{-1}x\right]dx=\underset{0}{\overset{\tan 1}{\int }}0dx+\underset{\tan 1}{\overset{102}{\int }}1dx$
$={\left(x\right)}_{\tan 1}^{102}=102-\tan 1$