Evaluate the following integrals: 1 5x^2-2x dx - Vidyadip

Question

Evaluate the following integrals:$\int\frac{1}{\sqrt{5\text{x}^2-2\text{x}}}\text{ dx}$

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Answer

$\int\frac{\text{dx}}{\sqrt{5\text{x}^2-2\text{x}}}$
$=\int\frac{\text{dx}}{\sqrt{5\big(\text{x}^2-\frac{2}{5}\text{x}\big)}}$
$=\frac{1}{\sqrt5}\int\frac{\text{dx}}{\sqrt{\text{x}^2-\frac{2}{5}\text{x}+\big(\frac{1}{5}\big)^2-\big(\frac{1}{5}\big)^2}}$
$=\frac{1}{\sqrt5}\int\frac{\text{dx}}{\big(\text{x}-\frac{1}{5}\big)^2-\big(\frac{1}{5}\big)^2}$
$=\frac{1}{\sqrt5}\log\bigg|\text{x}-\frac{1}{5}+\sqrt{\big(\text{x}-\frac{1}{5}\big)^2+\big(\frac{1}{5}\big)^2}\bigg|+\text{C}$
$=\frac{1}{\sqrt5}\log\Big|\frac{5\text{x}-1}{5}+\frac{\sqrt{5\text{x}^2-2\text{x}}}{\sqrt5}\Big|+\text{C}$

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