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If int(0)^(x)f(x)sint dt=" constant, " 0...

If `int_(0)^(x)f(x)sint dt=" constant, " 0 lt x lt 2pi and f(pi)=2`, then the value of `f(pi//2)` is

A

3

B

2

C

4

D

8

Text Solution

Verified by Experts

The correct Answer is:
C
`int_(0)^(x)f(x) sin t dt = "constant"`
Differentiate both side w.r.t. x
`f'(x)(1-cosx)+f(x) sin x = 0`
`rArr" "int(f'(x))/(f(x))dx=int(sinx)/(cosx-1)dx`
`rArr" "ln|f(x)|=-2 ln sin.(x)/(2)+lnc`
`rArr" "f(x)=(c)/((sin.(x)/(2))^(2))`
`f(pi)=2rArrc=2`
`rArr" "f((pi)/(2))=4`
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