Evaluate: `int_(sqrt(2))^(sqrt(2)+1)((x^2-1))/((x^2+1)^2)dx`

Answer

`0`

Explanation

`I=int_(sqrt(2)-1)^(sqrt(2)+1)((x^(2)-1))/((x^(2)+1)^(2))dx=int_(1//a)^(a)((x^(2)-1))/((x^(2)+1)^(2))dx`, where `a=sqrt(2)+1`
Put `x=1/t`. So `dx=-1/(t^(2))dt`
`:.I=int_(a)^(1//a)(1/(t^(2))-1)/((1/(t^(2))+1)^(2))-(-1/(t^(2)))dt`
`=-int_(a)^(1//a)((1-t^(2))t^(4))/(t^(4)(1+t^(2))^(2))dt`
`=-int_(a)^(1//a)((1-t^(2)))/((1+t^(2))^(2))dt`
`=int_(a)^(1//a)(t^(2)-1)/((t^(2)+1)^(2))dt`
`=-int_(1//a)^(a)(t^(2)-1)/((t^(2)+1)^(2))dt=-1`
or `2I=0`
or `I=0`