Evaluate the following integrals:


Evaluate the following integrals:

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


We have $\int \frac{\mathrm{dx}}{\sqrt{5 \mathrm{x}^{2}-2 \mathrm{x}}}=\int \frac{\mathrm{dx}}{\sqrt{5\left(\mathrm{x}^{2}-\frac{2 \mathrm{x}}{5}\right)}}$

$=\frac{1}{\sqrt{5}} \int \frac{d x}{\sqrt{\left(x-\frac{1}{5}\right)^{2}-\left(\frac{1}{5}\right)^{2}}}$ completing the square

Put $x-1 / 5=t$ then $d x=d t$

Therefore $\int \frac{d x}{\sqrt{5 x^{2}-2 x}}=\frac{1}{\sqrt{5}} \int \frac{d x}{\sqrt{(t)^{2}-\left(\frac{1}{5}\right)^{2}}}$

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

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

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