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If I(m)=int(0)^(pi)log(e)(1-2mcosx+m^(2)...

If `I(m)=int_(0)^(pi)log_(e)(1-2mcosx+m^(2))dx`, Then find the value of I(81/9)

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The correct Answer is:
2
`I(m)=int_(0)^(pi)log_(e)(1-2mcosx+m^(2))dx`
`impliesI(-m)=int_(0)^(pi)log_(e)(1+2mcosx+m^(2))dx`
`=int_(0)^(pi)log_(e)(1+2mcos(pi-x)+m^(2))dx`
`=int_(0)^(pi)log_(e)(1-2mcosx+m^(2))dx`
`=I(m)=I(m)`
`impliesI(m)+I(-m)=int_(0)^(pi)log_(e)(1-2m^(2)cos2x+m^(4))dx`
put `2x=t`
`impliesI(m)+(-m)=I(m^(2))`
`implies2I(m)=I(m^(2))`
`implies(I(m^(2)))/(I(m))=2`
`=(I(9))/(I(3))=2` and `(I(25))/(I(5))=2`
`implies(I(81))/(I(91))=2` and `(I(9))/(I(3))=2`
`implies (I(81))/(I(3))=4`
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