Evaluate the following integral:`int_0^oox/((1+x)(1+x^2))dx`

Explanation

Given:`int_0^oox/((1+x)(1+x^2))dx`

`I=int_0^oox/((1+x)(1+x^2))dx`

`x/((1+x)(1+x^2))=A/(1+x)+(Bx+C)/(x^2+1)`

`implies x = A ( x^2 + 1 ) + ( 1 + x ) ( B x + C )`

`impliesx=A(x^2+1)+(Bx+Bx^2+C+Cx)`

`implies x=(A+B)x^2+(B+C)x+(A+C)`

On comparing both sides , we get

` A + B = 0 , B + C = 1 , A + C = 0 A+B=0,B+C=1,A+C=0 ... (i)`

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