Evaluate: ∫ (x-1)/(√(x²-x)) dx.

Write $x - 1 = \frac12(2x - 1) - \frac12$. $I = \frac12\int\frac{2x-1}{\sqrt{x^2-x}}dx - \frac12\int\frac{dx}{\sqrt{x^2-x}}$ First integral: put $x^2 - x = t$, giving $\frac12\cdot2\sqrt{x^2-x} = \sqrt{x^2-x}$. Second: $x^2 - x = \left(x-\frac12\right)^2 - \left(\frac12\right)^2$, so $\int\frac{dx}{\sqrt{x^2-x}} = \log\left|x - \frac12 + \sqrt{x^2-x}\right|$. Hence $I = \sqrt{x^2-x} - \frac12\log\left|x - \frac12 + \sqrt{x^2-x}\right| + C$.