Find the integral: d x 5 x^ 2 -2 x - Vidyadip

Question

Find the integral: $\int \frac{d x}{\sqrt{5 x^{2}-2 x}}$

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Answer

We have $\int \frac{d x}{\sqrt{5 x^{2}-2 x}}=\int \frac{d x}{\sqrt{5\left(x^{2}-\frac{2 x}{5}\right)}}$
= $\frac{1}{\sqrt{5}} \int \frac{d x}{\sqrt{\left(x-\frac{1}{5}\right)^{2}-\left(\frac{1}{5}\right)^{2}}}$ (completing the square)
Put $x-\frac{1}{5}=t$. Then dx = dt
Therefore, $\int \frac{d x}{\sqrt{5 x^{2}-2 x}}=\frac{1}{\sqrt{5}} \int \frac{d t}{\sqrt{t^{2}-\left(\frac{1}{5}\right)^{2}}}$
= $\frac{1}{\sqrt{5}} \log |t+\sqrt{t^{2}-\left(\frac{1}{5}\right)^{2}}|+C$
= $\frac{1}{\sqrt{5}} \log \left|x-\frac{1}{5}+\sqrt{x^{2}-\frac{2 x}{5}}\right|+C$

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