Question

Integrate the function `1/sqrt((x - a)(x - b))`

Answer 1

Let `I = int dx/sqrt((x - a) (x - b))`

`= int dx/sqrt(x^2 - (a + b)x + ab)`

`= int dx/{[x^2 - 2  ((a + b)/2)  x + ((a + b)/2)^2] + ab - ((a + b)/2)^2`

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

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

`= log [x - (a + b)/2 + sqrt((x - a)(x - b))] + C`