Use \(\int_0^\pi g(x)\,dx = \int_0^\pi g(\pi - x)\,dx\):

\(I = \int_0^\pi [\cot x]\,dx = \int_0^\pi [-\cot x]\,dx\)

Add them: \(2I = \int_0^\pi \left([\cot x] + [-\cot x]\right)dx\).

For non-integer \(t\), \([t] + [-t] = -1\). (\(\cot x\) is an integer only at isolated points, which don't affect the integral.)

So \(2I = -\pi\), and \(I = -\frac{\pi}{2}\).

Answer: (D)