Question

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

Answer 1

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

$\int \frac{dx}{\sqrt{5\left[x^2-\frac{2}{5}x\right]}}$

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

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

$=\frac{1}{\sqrt{5}}\times \log \left|x-\frac{1}{5}+\sqrt{{\left(x-\frac{1}{5}\right)}^2-{\left(\frac{1}{5}\right)}^2}\right|+c$

$=\frac{1}{\sqrt{5}}\log \left|x-\frac{1}{5}+\sqrt{x^2-\frac{2}{5}x}\right|+c$