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If `f(x)` is a function satisfying `f(1/x)+x^2f(x)=0` for all nonzero `x` , then evaluate `int_(sintheta)^(cos e ctheta)f(x)dx`

A

`sin theta+"cosec" theta`

B

`sin^(2)theta`

C

`"cosec" ^(2)theta`

D

none of these

Text Solution

Verified by Experts

We have,
`f((1)/(x))+x^(2)f(x)=0 rArr r(x)=-(1)/(x^(2))f((1)/(x))`
`:. l=underset(sin theta)overset("cosec "theta)int f(x)dxunderset(sin theta)overset("cosec "theta)int-(1)/(x^(2))f((1)/(x))dx=underset("cosec" theta)overset(sin theta)intf(t)dt" where " t=(1)/(x)`
`rArr I=-underset(sin theta)overset("cosec "theta)int f(t)dt=-I rArr 2I=0 rArr I=0`
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