Question
$\int_{0}^{\pi}[\cot x] d x$, जहाँ $[.]$ महत्तम पूर्णांक फलन (greatest integer function) को दर्शाता है, बराबर है
$I = \int\limits_0^\pi {\left[ {\cot \left( {\pi - x} \right)} \right]} dx$
$ = \int\limits_0^\pi {\left[ { - \cot x} \right]} dx$
Adding we have
$2I = \int\limits_0^\pi {\left\{ {\left[ {\cot x} \right] + \left[ { - \cot x} \right]} \right\}} dx$
$2I = \int\limits_0^\pi {\left( { - 1} \right)} dx = - \pi $
$\therefore I = - \pi /2$
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