Question

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

Answer 1

We have,

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

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

Making perfect square and we get,

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

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

Using formula,

$\Rightarrow \int \frac{dx}{\sqrt{x^2-a^2}}=\ln |x+\sqrt{x^2-a^2}|$

Therefore,

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

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

Hence, this is the answer.