Compute the integrals: `int_0^oof(x^n+x^(-n))logx(dx)/(1+x^2)`

Explanation

Let `I=int_(0)^(oo) fx^(n)+x^(-n)In x (dx)/(1+x^(2))`
Let `t=1//x` or `x=1//t`. So `dx=-1/(t^(2))dt`.
Also when `xto0,t to oo`, when `xto oo, t to 0`. Thus,
`I=int_(0)^(oo) f(x^(n)+x^(-n)) In x (dx)/(1+x^(2))`
`=int_(oo)^(0)f(t^(-n)+t^(n))In (1/t)(-(dt)/(t^(2)))/(1+1/(t^(2)))`
`=-int_(0)^(oo) f(t^(n)+t^(-n))In (t)(dt)/(1+t^(2))`
`=-I`
or `2I=0` or `I=0`