Prove that `int_0^x[cot^(-1)x]dx ,w h e r e[dot]` denotes the greatest integer function.

Answer

`pi+cot1+cot2`

Explanation

We have `int_(-pi//2)^(2pi)[cot^(-1)x]dx`
We have that `cot^(-1)xepsilon[0,pi]`
so `[cot^(-1)x]=0` for `cot^(-1)x epsilon(0,1)` or `xepsilon(cot1,oo)`
`[cot^(-1)x]=1` for `cot^(-1)xepsilon[1,2)` or `xepsilon(cot2, cot1]`
`[cot^(-1)x=2` for `cot^(-1)x epsilon[2,3)` or `x epsilon(cot3, cot2]`
`[cot^(-1)x]=3` for `cot^(-1)xepsilon[3,pi)` or `x epsilon(-oo,cot3]`
`:. int_(cot1)^(2pi) 0dx+int_(cot2)^(cot1) 1dx+int_(-pi//2)^(cot2) 2dx`
`=cot1-cot2+2cot2+pi`
`=cot1+cot2+pi`