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

Question

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

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Answer

Let $\text{I}=\int\sqrt{\text{x}^2-2\text{x}}\text{dx}$
$\Rightarrow\text{I}=\int\sqrt{\text{x}^2-2\text{x}+1-1}\text{dx}$
$\Rightarrow\text{I}=\int\sqrt{(\text{x}-1)^2-1^2}\text{dx}$
$\because\ \int\sqrt{\text{x}^2-\text{a}^2}\text{dx}=\frac{\text{x}}{2}\sqrt{\text{x}^2-\text{a}^2}-\frac{\text{a}^2}{2}\ln\big(|\text{x}+\sqrt{\text{x}^2-\text{a}^2}|\big)+\text{C}$
$\therefore\ \text{I}=\frac{(\text{x}-1)}{2}\sqrt{(\text{x}-1)^2-1}-\frac{1}{2}\ln\big|(\text{x}-1)+\sqrt{\text{x}^2-2\text{x}}\big|+\text{C}$

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