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int(0)^(pi)[cotx]dx, where [.] denotes t...

`int_(0)^(pi)[cotx]dx,` where [.] denotes the greatest integer function, is equal to

A

`(pi)/2`

B

`1`

C

`-1`

D

`-(pi)/2`

Text Solution

Verified by Experts

The correct Answer is:
D
Let `I=int_(0)^(pi)[cotx]dx` …………..i
`=int_(0)^(pi)[cot(pi-x)]dx=int_(0)^(pi)[-cotx]dx`………….ii
Adding i and ii we get
`2I=int_(0)^(pi)[cotx] dx+int_(0)^(pi)[-cotx]dx=int_(0)^(pi)(-1)dx`
[since `[x]+[-x]` is equal to `-1` if `x !inZ` and is equal to 0 if `x epsilonZ`]
`=[-x]_(0)^(pi)=-pi`
`:.I=-(pi)/2`
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