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The value of int(0)^(oo)(logx)/(a^(2)+x^...

The value of `int_(0)^(oo)(logx)/(a^(2)+x^(2))dx` is

A

`(2piloga)/(a)`

B

`(pi log a)/(2a)`

C

`pi loga`

D

0

Text Solution

Verified by Experts

The correct Answer is:
B
`I=int_(0)^(oo)(logx)/(a^(2)+x^(2))dx`
Put `x=(a^(2))/(y),dx=-(a^(2))/(y^(2))dy`
`therefore" "I=int_(oo)^(0)(log((a^(2))/(y)))/(a^(2)+(a^(4))/(y^(2)))((-a^(2))/(y^(2)))dy`
`" "=int_(oo)^(0)((loga^(2)-logy))/(a^(2)+y^(2))(-dy)`
`" "=log(a^(2))int_(0)^(oo)(dy)/(a^(2)+y^(2))-int_(0)^(oo)(logy)/(a^(2)+y^(2))dy`
`" "log(a^(2))(1)/(a)(tan^(-1)((y)/(a)))_(0)^(oo)-I`
`rArr" "2I=(2loga)/(a).(pi)/(2)`
`rArr" "I=(piloga)/(2a)`
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