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Question
Integrate the following with respect to x.
`1/(x + sqrt(x^2 - 1)`
Solution
`int 1/(x + sqrt(x^2 - 1)) "d"x`
By rationalisation
= `int (1 xx (x - sqrt(x^2 - 1)))/((x + sqrt(x^2 - 1)) xx (x - sqrt(x^2 - 1))) "d"x`
= `int ((x - sqrt(x^2 - 1)))/((x^2) - (sqrt(x^2 - 1)^2)) "d"x`
= `int ((x - sqrt(x^2 - 1)))/(x^2 - (x^2 - 1)) "d"x`
= `int (x - sqrt(x^2 - 1))/(x^2 - x^2 + 1) "d"x`
= `int (x - sqrt(x^2 - 1)) "d"x`
= `int x "d"x - int sqrt(x^2 - 1) "d"x`
= `x^2/2 - {x/2 sqrt(x^2 - 1) - (1)^2/2 log|x + sqrt(x^2 - 1)|} + "c"`
= `x^2/2 - x/2 sqrt(x^2 - 1) + 1/2 log|x + sqrt(x^2 - 1)| + "c"`
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