Evaluate: `int_1^oo(e^(x+1)+e^(3-x))^(-1)dx`

Answer

`(pi)/(4e^(2))`

Explanation

`I=int_(1)^(oo)(dx)/((ee^(x)+e^(3)e^(-x)))`
`=int_(1)^(oo) (e^(x)dx)/(e(e^(2x)+e^(2)))` (multiply `N^(R)` and `D^(r)`by `e^(x)`)
Put `e^(x)=t` or `e^(x) dx=dt`
`:. I=1/e int_(e)^(oo) (dt)/(t^(2)+e^(2))`
`=1/(e^(2))"tan"(-1)t/e|_(e)^(oo)`
`=1/(e^(2))[(pi)/2-(pi)/4]=(pi)/(4e^(2))`