int0^x[sint]dt ,w h e r ex in (2npi,(2n+...

`int_0^x[sint]dt ,w h e r ex in (2npi,(2n+1)pi),n in N ,a n d[dot]` denotes the greatest integer function is equal to `-npi` (b) `-(n+1)pi` `2npi` (d) `-(2n+1)pi`

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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