Find the value of `int_(1/2)^(2)e^(|x-1/x|)dx`.

Answer

`esqrt(e)-1`

Explanation

Let `I=int_(1/2)^(2)e^(|x-1/x|)dx`…………….1
Put `x=1/t`,
`:. I=-int_(0)^(1/2)e^(|t- 1/t|)((dt)/(t^(2)))=int_(1/2)^(2)e^(|x-1/x|)(dx)/(x^(2))`……………2
Adding 1 and 2 we get
`2I=int_(1/2)^(2)e^(|x - 1/x|),(1+1/(x^(2)))dx`
`=int_(1/2)^(1)e^(-(x-1/x)),(1+1/(x^(2)))dx+int_(1)^(2)e^((x-1/x)),(1+1/(x^(2)))dx`
`=[-e^(-(x-1/x))]_(1//2)^(1)+[e^((x-1/x))]_(1)^(2)`
`=-1+e^(3//2)+e^(3//2)-1`
`=2(esqrt(e)-1)`
`impliesI=esqrt(e)-1`