Question

`int_0^1 "dx"/(sqrt(1 + x) - sqrtx)` = ?

Options

• `(2sqrt2)/3`

• `(4sqrt2)/3`

• `(8sqrt2)/3`

• `(sqrt2)/3`

Answer 1

`(4sqrt2)/3`

Explanation:

`int_0^1 "dx"/(sqrt(1 + x) - sqrtx)`

`= int_0^1 ((sqrt(1 + x) + sqrtx))/((sqrt(1 + x) - sqrtx)(sqrt(1 + x) + sqrtx))`dx

`= int_0^1 ((sqrt(1 + x) + sqrtx))/(1 + x - x)`dx

`= int_0^1 sqrt(1 + x)  "dx" + int_0^1 sqrtx  "dx"`

`= [2/3 (1 + x)^(3/2)]_0^1 + [2/3 (x)^(3/2)]_0^1`

`= (4sqrt2)/3 - 2/3 + 2/3 - 0`

`= (4sqrt2)/3`