Question

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

Answer 1

Let `I = (x + 2)/sqrt(x^2 - 1)  dx`

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

`= I_1 + I_2 + C` (Say)                        ....(i)

Now `I_1 = int x/sqrt(x^2 - 1)  dx`

Put x² - 1 = t

2x dx = dt

`= 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`                   ....(ii)

and `I_2 = int 2/sqrt(x^2 - 1)  dx`

`= 2 log abs (x + sqrt(x^2 - 1) + C_2`                   .....(iii)

From (i), (ii) and (iii), we get,

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