Question

$\int \frac{1}{\sqrt{5x^2-2x}}dx$

Answer 1

$$\begin{gathered} \int \frac{dx}{\sqrt{5x^2-2x}} \\ =\int \frac{dx}{\sqrt{5\left(x^2-{\displaystyle \frac{2}{5}}x\right)}} \\ =\frac{1}{\sqrt{5}}\int \frac{dx}{\sqrt{x^2-{\displaystyle \frac{2}{5}}x+{\left({\displaystyle \frac{1}{5}}\right)}^2-{\left({\displaystyle \frac{1}{5}}\right)}^2}} \\ =\frac{1}{\sqrt{5}}\int \frac{dx}{\sqrt{{\left(x-{\displaystyle \frac{1}{5}}\right)}^2-{\left({\displaystyle \frac{1}{5}}\right)}^2}} \\ =\frac{1}{\sqrt{5}}\log \left|x-\frac{1}{5}+\sqrt{{\left(x-\frac{1}{5}\right)}^2+{\left(\frac{1}{5}\right)}^2}\right|+C \\ =\frac{1}{\sqrt{5}}\log \left|\frac{5x-1}{5}+\frac{\sqrt{5x^2-2x}}{\sqrt{5}}\right|+C \end{gathered}$$