Integrate the rational function 1-x^ 2 x(1-2 x)

Question

Integrate the rational function $\frac{1-x^{2}}{x(1-2 x)}$

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Answer

On dividing $1 - x^2$ by $x(1 - 2x),$
we get,$\frac{1-x^{2}}{x(1-2 x)}=\frac{1}{2}+\frac{1}{2}\left(\frac{2-x}{x(1-2 x)}\right).......(i)$
Now, let $\frac{2-x}{x(1-2 x)}=\frac{A}{x}+\frac{B}{(1-2 x)}$
$(2 - x) = A(1 - 2x) + Bx …...(ii)$
Now, substituting $x = 0$ and $\frac{1}{2}$ in equation $(ii),$ we get,
$A = 2$ and $B = 3$
Thus, $\frac{2-x}{x(1-2 x)}=\frac{2}{x}+\frac{3}{(1-2 x)}$
Now, putting this value in equation $(ii),$ we get,
$\frac{1-x^{2}}{x(1-2 x)}=\frac{1}{2}+\frac{1}{2}\left(\frac{2}{x}+\frac{3}{(1-2 x)}\right)$
$\Rightarrow$$\int \frac{1-x^{2}}{x(1-2 x)} d x=\int\left\{\frac{1}{2}+\frac{1}{2}\left(\frac{2}{x}+\frac{3}{(1-2 x)}\right)\right\} d x$
$= \frac{x}{2}+\log |x|+\frac{3}{2(-2)} \log |1-2 x|+C$
$= \frac{x}{2}+\log |x|-\frac{3}{4} \log |1-2 x|+C$

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