Evaluate: 1 - x^ 2 x(1 - 2x) dx. - Vidyadip

Question

Evaluate:$\int\frac{1 - x^{2}}{x(1 - 2x)}\text{dx}$.

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Answer

$\text{I}=\int\frac{1 - x^{2}}{x(1-2x)}\text{dx}=\frac{1}{2}\int\frac{2-2x^{2}}{2-2x^{2}}\text{dx}$
$=\frac{1}{2}\int\Bigg[1+\frac{2-x}{x(1-2x)}\Bigg]\text{dx}$
$=\frac{\text{x}}{2} +\frac{1}{2}\int\frac{\text{2-x}}{\text{x(1-2x)}}\text{dx}$
Let $\frac{\text{2 - x}}{\text{x(1 - 2x)}}=\frac{\text{A}}{\text{x}}+\frac{\text{B}}{\text{1 - 2x}}\cdot\text{Getting A = 2, B = 3}$
$\therefore\text{I}=\frac{\text{x}}{2}+\frac{1}{2}\int\Bigg(\frac{2}{\text{x}}+\frac{\text{3}}{\text{1 - 2x}}\Bigg)\text{dx}$
$=\frac{\text{x}}{2}+\log|\text{x}|-\frac{3}{4}\log|1-2\text{x}|+\text{c}.$

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