Question

`int dx/(sqrt(x + 3) - sqrt(x + 2))` = ______ 

Options

• `2/3log|sqrt(x + 3) - sqrt(x + 2)| + c`

• `log|sqrt(x + 3) - sqrt(x + 2)| + c`

• `2/3[(x + 3)^{3/2} + (x + 2)^{3/2}] + c`

• `(x + 3)^{3/2} + (x + 2)^{2/3} + c`

Answer 1

`int dx/(sqrt(x + 3) - sqrt(x + 2))` = `underline(2/3[(x + 3)^{3/2} + (x + 2)^{3/2}] + c)`

Explanation:

Rationalizing the denominator, we get

`int dx/(sqrt(x + 3) - sqrt(x + 2))`

= `int(sqrt(x + 3) + sqrt(x + 2))/((sqrt(x + 3) - sqrt(x + 2))(sqrt(x + 3) + sqrt(x + 2))) dx`

= `int (sqrt(x + 3) + sqrt(x + 2))/((x + 3) - (x + 2)) dx = int(sqrt(x + 3) + sqrt(x + 2))/(x + 3 - x - 2)dx`

= `int {(x + 3)^{1/2} + (x + 2)^{1/2}} dx`

= `(x + 3)^{3/2}/(3/2) + (x + 2)^{3/2}/(3/2) + c`

= `2/3[(x + 3)^{3/2} + (x + 2)^{3/2} + c]`