The value of the definite integral `int_(2-1)^(sqrt(2)+1)(x^4+x^2+2)/((x^2+1)^2)dxe q u a l s___`

Answer

2

Explanation

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