Question

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

Answer 1

\[\int\frac{dx}{\sqrt{5 x^2 - 2x}}\]
\[ = \int\frac{dx}{\sqrt{5\left( x^2 - \frac{2}{5}x \right)}}\]
\[ = \frac{1}{\sqrt{5}}\int\frac{dx}{\sqrt{x^2 - \frac{2}{5}x + \left( \frac{1}{5} \right)^2 - \left( \frac{1}{5} \right)^2}}\]
\[ = \frac{1}{\sqrt{5}}\int\frac{dx}{\sqrt{\left( x - \frac{1}{5} \right)^2 - \left( \frac{1}{5} \right)^2}}\]
\[ = \frac{1}{\sqrt{5}} \text{ log }\left| x - \frac{1}{5} + \sqrt{\left( x - \frac{1}{5} \right)^2 + \left( \frac{1}{5} \right)^2} \right| + C\]
\[ = \frac{1}{\sqrt{5}} \text{ log }\left| \frac{5x - 1}{5} + \frac{\sqrt{5 x^2 - 2x}}{\sqrt{5}} \right| + C\]