Question

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

Answer 1

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

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

Adding & subtracting $\frac{1}{25}$.

$\Rightarrow \frac{1}{\sqrt{5}}$ $\int \frac{dx}{\sqrt{x^2-\frac{2}{5}x+\frac{1}{25}-\frac{1}{25}}}$

$=\frac{1}{\sqrt{5}}\frac{dx}{\sqrt{{(x-\frac{1}{5})}^2-{\left(\frac{1}{5}\right)}^2}}$

$=\frac{1}{\sqrt{5}}log|(x-\frac{1}{5})+\sqrt{x^2-\frac{2}{5}x}|+C$.