$\int_{0}^{\pi}[\cot x] dx$, $[\bullet]$ denotes the greatest integer function, is equal to

JEE Main · 2009 · 12 Apr 2009 · Offline · Definite Integrals

\int_{0}^{π} [cot x] d x

[∙]

denotes the greatest integer function, is equal to

Answer: (D)

Let

I = \int_{0}^{π} [cot x] d x

and replace

x

π - x

to get the companion form

I = \int_{0}^{π} [- cot x] d x

, since

cot (π - x) = - cot x

. Adding the two,

2 I = \int_{0}^{π} ([cot x] + [- cot x]) d x

, and because

[y] + [- y] = - 1

whenever

y

is not an integer, which is true for

cot x

almost everywhere on

(0, π)

, this is

\int_{0}^{π} (- 1) d x = - π

. Hence

I = - \frac{π}{2}

JEE Main · 2009 · 12 Apr 2009 · Offline · Single correct option

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