Question

$\int \sqrt{x^2-2x}dx$

Answer 1

$I=\int \sqrt{x^2-2x}dx$

$$\begin{gathered} \Rightarrow I=\int \sqrt{x^2-2x+1-1}dx \\ \Rightarrow I=\int \sqrt{(x-1{)}^2-1^2}dx \\ \because \int \sqrt{x^2-a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\ln \left(\left|x+\sqrt{x^2-a^2}\right|\right)+c \\ \therefore I=\frac{(x-1)}{2}\sqrt{(x-1{)}^2-1}-\frac{1}{2}\ln \left|\left(x-1\right)+\sqrt{x^2-2x}\right|+c \end{gathered}$$