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Question
Evaluate the following integral:
`int 1/(sqrt(x + 2) - sqrt(x + 3)) "d"x`
Solution
`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"`
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