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Evaluate: int0^(2pi)[sinx]dx ,w h e r e[...

Evaluate: `int_0^(2pi)[sinx]dx ,w h e r e[dot]` denotes the greatest integer function.

A

`4n-cosx`

B

`4n-sinx`

C

`4n+1-cosx`

D

`4n-1-cosx`

Text Solution

Verified by Experts

The correct Answer is:
C
`int_(0)^(x)|sint|dt=int_(0)^(2npi)|sint|dt+int_(2npi)^(x)|sint|dt`
`=2nint_(0)^(pi) |sint|dt+int_(2npi)^(x)sin tdt`
(as `x` lies in either first or second quadrant)
`=2n(-cost)_(0)^(pi)+(-cost)_(2npi)^(x)=4n-cosx+1`
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