Question

Integrate the function `(x - 1)/sqrt(x^2 - 1)`

Answer 1

Let  `I = int (x - 1)/sqrt(x^2 - 1)  dx`

`= int x/(sqrt(x^2 - 1))  dx - int 1/sqrt(x^2 - 1)  dx`

`= I_1 - I_2`  (say)

Now , `I_1 = x/sqrt(x^2 - 1)  dx`

Put x² - 1 = t

2x dx = dt ⇒ x dx = `1/2` dt

`therefore I = 1/2 int dt/sqrtt = 1/2 int t^((-1)/2) dt`

`= 1/2 xx t^(1/2)/(1/2) + C_1 = sqrtt = C_1`

`= sqrt(x^2 - 1) + C_1`

and `I_2 int 1/sqrt(x^2 - 1)  dx`

`= log [x + sqrt(x^2) - 1] + C_2`            `....[∵ int dx/sqrt(x^2 - a^2) = log |x + sqrt(x^2 - a^2)| + C]`

`therefore I = sqrt(x^2 - 1) - log |x + sqrt(x^2 - 1)| +C`