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Question
Integrate the following with respect to x.
`1/(sqrt(x + 1) + sqrt(x - 1))`
Solution
`int (1/(sqrt(x + 1) + sqrt(x - 1))) "d"x`
= `int (1 xx (sqrt(x + 1) - sqrt(x - 1)))/((sqrt(x + 1) + sqrt(x - 1)) xx (sqrt(x + 1) - sqrt(x - 1))) "d"x`
= `int ((sqrt(x + 1) - sqrt(x - 1)))/((x + 1) - (x + 1)) "d"x`
= `int ((x + 1)^(1/2) - (x - 1)^(1/2))/(x + 1 - x + 1) "d"x`
= `1/2 [(x + 1)^(1/2 + 1)/((1/2 + 1)) - (x - 1)^(1/2 + 1)/((1/2 + 1))] + "c"`
= `1/2 [(x + 1)^(3/2)/((3/2)) - (x - 1)^(3/2)/((3/2))] + "c"`
= `1/2 [2/3 (x + 1)^(3/2) - 2/3 (x - 1)^(3/2)] + "c"`
= `2/2 [1/3 (x + 1)^(3/2) - 1/3 (x - 1)^(3/2)] + "c"`
= `1/2 [(x + 1)^(3/2) - (x - 1)^(3/2)] + "c"`
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