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Prove that: int(0)^(oo) (x)/((1+x)(1+x^(...

Prove that: `int_(0)^(oo) (x)/((1+x)(1+x^(2)))dx =(pi)/(4)`

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Let x=tan 0
`rArr " " dx =sec^(2) 0 d0`
`At x=0 , " "tan 0=0 " " rArr " "0=0`
`At x=oo , " "tan 0 =oo " "rArr " "0=(pi)/(2)`
`"Let I "=int_(0)^(oo) (x) /((1+x)(1+x^(2)))dx`
`=int_(0)^(pi//2) (tan0)/((1+tan 0)(1+tan^(2)0)) .sec^(2) 0 d0`
`rArr I= int_(0)^(pi//2) (tan0)/(1+tan 0) d0`
`rArr I=int_(0)^(pi//2) (tan.((pi)/2-0))/(1+tan .((pi)/(2)-0))do`
[From Property (4)]
`=int_(0)^(pi//2) (cot0)/(1+cot 0) d0`
` =int_(0)^(pi//2) (1//tan0)/(1+(1)/(tan0))d0`
`rArr " "I= int_(0)^(pi//2) (1)/(tan0+1)d0......(2)`
Adding eqs. (1) and(2)
`2I=int_(0)^(pi//2) (tan0+1)/(tan0+1)d0`
`=int_(0)^(pi//2) 1d0`
` =[0]_(0)^(pi//2)`
`=pi//2`
`rArr " "i=(pi)/(4)`Hence Proved.
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