Evaluate: `int_(pi/2)^(2pi)[cot^(-1)]dx` , where `[dot]` denotes the greatest integer function

Prove that int_(-(pi)/(2))^(2 pi)[cot^(-1)x]dx, where [.] denotes the greatest integer function.

The value of int_(-pi//2)^(pi//2)[ cot^(-1)x] dx (where ,[.] denotes greatest integer function) is equal to

int_(0)^(pi)[cotx]dx, where [.] denotes the greatest integer function, is equal to

The value of int_(0)^(2pi)[sin 2x(1+cos 3x)]dx , where [t] denotes the greatest integer function, is:

Evaluate: int_(0)^(2 pi)[sin x]dx, where [.] denotes the greatest integer function.

Evaluate: int_(0)^((5 pi)/(12))[tan x]dx, where [.] denotes the greatest integer function.

The value of int_(1)^(10pi)([sec^(-1)x]) dx (where ,[.] denotes the greatest integer function ) is equal to