`int_- 1^1(e^(-1/ x))/(x^2(1+e^(-2/ x)))dx` is equal to :

Options

A. `(pi)/2=2tan^(-1)e`
B. `(pi)/2-2cot^(-1)e`
C. `2tan^(-1)e`
D. `pi-2tan^(-1)e`

Answer

D

Explanation

`I=int_(-1)^(1)(e^(-1//x))/(x^(2)(1+e^(-2//x)))dx=2 int_(0)^(1)(e^(-1//x))/(x^(2)(1+e^(-2//x)))dx`
Put `e^(-1/x)=t`
`implies 1/(x^(2))e^(-1/x)dx=dt`
`:. I=2 int_(0)^(1//2) (dt)/(1+t^(2))`
`=2[tan^(-1)t]_(0)^(1//e)`
`=2"tan"^(-1)1/e`
`=2[(pi)/2-tan^(-1)e]`
`=pi-2 tan^(-1) e`