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Evaluate the definite integral I = integral from 0 to pi of [cot(x)] dx, where [.] denotes the greatest integer function (floor function). Find the value of [-I].

1. 0
2. 1
3. -1
4. 2

Correct answer: 1

Solution

Using the substitution x -> (pi - x): integral from 0 to pi [cot(pi - x)] dx = integral from 0 to pi [-cot x] dx. So I = integral[cot x]dx and J = integral[-cot x]dx. I + J = integral ([cot x] + [-cot x]) dx. For non-integer values, [t] + [-t] = -1. Since cot x is never an integer for almost all x in (0, pi), I + J = -pi. Also by symmetry (substitution u = pi - x), J = I, so 2I = -pi => I = -pi/2. Therefore -I = pi/2 and [-I] = [pi/2] = [1.5707...] = 1.

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