Question

Evaluate the following integral:

`int 1/(sqrt(x + 2) - sqrt(x + 3))  "d"x`

Answer 1

`int 1/(sqrt(x + 2) - sqrt(x + 3))  "d"x`

Conjugating the Denominator

`int 1/sqrt(x + 2) - 1/sqrt(x + 3) xx ((sqrt(x + 2) + sqrt(x + 3))/(sqrt(x + 2) + sqrt(x + 3)))  "d"x`

= `int ((sqrt(x + 2) + sqrt(x + 3)))/((sqrt(x + 2))^2 + (sqrt(x + 3))^2)  "d"x`

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

= `int (sqrt(x + 2) + sqrt(x + 3))/((-1))  "d"x`

= `- [int (x + 2)^(1/2)  "d"x + int (x + 3)^(1/2)  "d"x] + "c"`

= `- [((x + 2)^(3/2))/((3/2)) + (x + 3)^(1/2)/((3/2))]+ "c"`

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