Question

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

Answer 1

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

`:. x - 1 = A  d/dx (x^2- x) + B`

`x - 1 = A(2x -1) + B`

`1 = 2A => A = 1/2`

`-1 = -A+B => -1 = (-1)/2 + B => B = (-1)/2`

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

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

`= 1/2 xx 2sqrt(x^2 - x) - 1/2 xx log|(x - 1/2) + sqrt((x- 1/2)^2 - (1/2)^2)| +C`

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