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A function f(x) satisfies f(x)=sinx+int0...

A function `f(x)` satisfies `f(x)=sinx+int_0^xf^(prime)(t)(2sint-sin^2t)dt` is

A

`f((pi)/(6))=1`

B

`g(x)=int_(0)^(x)f(t)` dt is increasing on `(0,pi)`

C

`f(0)=0`

D

f(x) is increasing on `(0,pi)`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
`f(x)=sin x +int_(0)^(x)f'(t)(2 sin t - sin^(2)t)dt`
`rArr" "f'(x)=cos x+f'(x)(2 sin x-sin^(2)x)`
`rArr" "(1-2 sin x+sin^(2)x)f'(x)=cosx`
`rArr" "f'(x)=(cosx)/((sin x-1)^(2))`
`rArr" "f(x)=(1)/((1-sinx))+c`
since f(0)= 0 we have `c=-1`
`rArr" "f(x)=(sinx)/(1-sinx)`
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