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Question
Integrate the following functions with respect to x:
`sqrt(x^2 - 2x - 3)`
Solution
`int sqrt(x^2 - 2x - 3) "d"x = int sqrt((x - 1)^2 - 1^2 - 3) "d"x`
= `int sqrt((x - 1)^2 - 4) "d"x`
= `int sqrt((x - 1)^2 - 2^2) "d"x`
Put x – 1 = t
dx = dt
= `int sqrt("t"^2 - 2^2) "dt"`
= `"t"/2 sqrt("t"^2 - 2^2) - 2^2/2 log |"t" + sqrt("t"^2 - 2^2)| + "c"`
= `(x - 1)/2 sqrt((x - 1)^2 - 4) - 2 log|x - 1 + sqrt((x - 1)^2 - 4)| + "c"`
= `(x - 1)/2 sqrt(x^2 - 2x + 1 - 4) - 2 log|x - 1 sqrt(x^2 - 2x + 1 - 4) | + "c"`
`int sqrt(x^2 - 2x - 3) "d"x = (x - 1)/2 sqrt(x^2 - 2x - 3) - 2 log |x - 1 + sqrt(x^2 - 2x - 3)| + "c"`
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