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