QJEEM09Ma8ad1: JEE Main 2009 Mathematics - Class 12 JEEM09Ma8ad1 Solution: where denotes the greatest integer function, is equal to:

Question

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\int\limits_0^\pi {\left[ {\cot x} \right]dx,}

where

\left[ . \right]

denotes the greatest integer function, is equal to:

Accepted Answer

Let

I = \int_0^\pi {\left[ {\cot x} \right]dx\,\,\,\,\,\,...\left( 1 \right)}

= \int_0^\pi {\left[ {\cot \left( {\pi - x} \right)} \right]} dx

= \int_0^\pi {\left[ { - \cot x} \right]dx\,\,\,\,\,\,...\left( 2 \right)}

Adding two values of

I

in

e{q^n}s\left( 1 \right)\,\,\&amp; \,\,\left( 2 \right),

We get

2I = \int_0^\pi {\left( {\left[ {\cot x} \right] + \left[ { - \cot x} \right]} \right)dx}

= \int_0^\pi {\left( { - 1} \right)dx}

\left[ {} \right.

As

\left[ x \right] + \left[ { - x} \right] = - 1,\,\,if\,\,

x \notin z

and

\left[ x \right] + \left[ { - x} \right] = 0,\,\,if\,\,x \in z

\left. {} \right]

= \left[ { - x} \right]_0^\pi = - \pi \Rightarrow I = - {\pi \over 2}

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