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`

Options

A. 0
B. 1
C. 2
D. 3

Answer

A

Explanation

`f((1)/(x))+x^(2)f(x)=0`
Let `I=int_(sintheta)^("cosec "theta)f(x)dx`
Put `x=(1)/(t)impliesdx=-(1)/(t^(2))dt`
`:." "I=-int_("cosec "theta)^(sintheta)f((1)/(t))(1)/(t^(2))dt`
`=int_(sintheta)^("cosec "theta)(1)/(x^(2))f((1)/(x))dx`
`impliesI=-int_(sintheta)^("cosec "theta)f(x)dximpliesI=0`