Find the integral of [tan^-1x]dx from 0 to 102, where [x] is the largest integer not exceeding x.

Put tan^-1[x] = y

or x=tany

or dx=sec^2ydy

Now integral becomes

0 to tan^(-1)(102) [y]sec^ydy

Split this integral as 0 to 1 + 1 to tan^(-1)(102)

from 0 to 1 [y]=0

from 1 to tan^(-1)(102), [y]=1

Hence, answer is 102-tan1