Question

`int_1^2 (1)/(x^2) e^(1/x) * dx` = ______.

Options

• `sqrt(e) + 1`

• `sqrt(e) - 1`

• `sqrt(e)(sqrt(e) - 1)`

• `(sqrt(e) - 1)/e`

Answer 1

`bb(sqrt(e)(sqrt(e) - 1))`

Explanation:

Let `t = 1/x`

⇒ `dt = - 1/x^2 dx`

Changing limits:

when x = 1, t = 1

when x = 2, t = `1/2`

So, `int_1^2 1/x^2 e^(1/x) dx = int_1^(1/2) e^t(-dt)`

= `int_(1/2)^1 e^t dt`

= `e^t|_(1//2)^1`

= `e - e^(1/2)`

= `sqrt(e) (sqrt(e) - 1)`