Question

Integrate the following functions with respect to x:

`sqrt(x^2 - 2x - 3)`

Answer 1

`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"`